Difference between revisions of "Find a Number Problems"

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(→‎Find One Number)
(→‎Find One Number)
 
(49 intermediate revisions by the same user not shown)
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== Find a Perfect Number ==
 +
 +
<div class="mw-collapsible mw-collapsed">perfect number<div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find a number[0] ]]
+
[[category: #find a number]]
−
[[comment: repeated subtraction]]
+
[[comment: perfect number]]
−
}}
+
}}</div></div>
  
 
== Find One Number ==
 
== Find One Number ==
  
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle37a=49</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: 37a=49]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle9a=1</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: 9a=1]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle13\sdot7\sdot3\sdot a=3</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: 13·7·3a=3]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{7a}{9}=49</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: 7a/9=45]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{a}{5}\sdot8=23</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: (5/a)·8=23]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{\frac{1}{5}a}{\sqrt{a}}=7</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: ⅕a/√a=7]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{3}{a}=5+\frac{7}{b}</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: 3/a=5+7/b]]
 +
}}</div></div>
 +
<br>
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(\frac{1}{3}\sdot a\right)+\left(\frac{1}{5}\sdot a\right)=10</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(\frac{1}{3}\sdot a\right)+\left(\frac{1}{5}\sdot a\right)=10</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
Line 36: Line 82:
 
}}</div></div>
 
}}</div></div>
 
<br>
 
<br>
−
 
+
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a+\left(\frac{1}{3}\sdot a\right)+\left(\frac{2}{5}\sdot a\right)=81</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: a+⅓a+⅖a=81]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a+\left[\left(2+\frac{2}{3}\right)\sdot a\right]=11</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: a+2⅔a=11]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(\frac{1}{2}\sdot a\right)+\left(\frac{1}{3}\sdot a\right)+\left(\frac{1}{4}\sdot a\right)-5=24</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: ½a+⅓a+¼a-5=24]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a+\frac{1}{2}\sdot\left(\frac{1}{4}\sdot a\right)+\left(\frac{3}{4}\sdot a\right)+\frac{1}{4}\sdot\left(\frac{1}{4}\sdot a\right)=2</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: a+½¼a+¾a+¼¼a=2]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a+\frac{1}{2}a+\frac{1}{5}a+\frac{1}{7}a=2</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: a+½a+⅕a+⅐a=2]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a+\frac{1}{2}\sdot\frac{1}{2}a+\frac{1}{3}\sdot\frac{1}{3}a+\frac{1}{5}\sdot\frac{1}{5}a=7</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: a+½½a+⅓⅓a+⅕⅕a=7]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(a+\frac{1}{3}a\right)+\left[\frac{2}{5}\sdot\left(a+\frac{1}{3}a\right)\right]+\left[\frac{3}{4}\sdot\left(a+\frac{1}{3}a\right)+\left[\frac{2}{5}\sdot\left(a+\frac{1}{3}a\right)\right]\right]=13</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: a+⅓a+⅖(a+⅓a)+¾(a+⅓a+⅖(a+⅓a))=13]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a-\left(\frac{1}{3}\sdot a\right)-\left(\frac{1}{4}\sdot a\right)=29</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: a-⅓a-¼a=29]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(\frac{1}{2}a+\frac{1}{3}a\right)-\left(\frac{1}{4}a+\frac{1}{5}a\right)=4</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: (½a+⅓a)-(¼a+⅕a)=4]]
 +
}}</div></div>
 +
<br>
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a-\left(\frac{1}{3}\sdot a\right)=\sqrt{a}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a-\left(\frac{1}{3}\sdot a\right)=\sqrt{a}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find one number]]
+
[[category: #find a number]]
 
[[comment: a-⅓a=√a]]
 
[[comment: a-⅓a=√a]]
 
}}</div></div>
 
}}</div></div>
Line 45: Line 144:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left[a-\left(\frac{2}{3}\sdot a\right)\right]^2=20</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left[a-\left(\frac{2}{3}\sdot a\right)\right]^2=20</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find one number]]
+
[[category: #find a number]]
 
[[comment: (a-⅓a)²=20]]
 
[[comment: (a-⅓a)²=20]]
 
}}</div></div>
 
}}</div></div>
Line 51: Line 150:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left[a\sdot\left(\frac{2}{3}\sdot a\right)\right]^2=36</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left[a\sdot\left(\frac{2}{3}\sdot a\right)\right]^2=36</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find one number]]
+
[[category: #find a number]]
 
[[comment: (a·⅔a)²=36]]
 
[[comment: (a·⅔a)²=36]]
 
}}</div></div>
 
}}</div></div>
Line 57: Line 156:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a^2\sdot\left(\frac{3}{4}\sdot a^2\right)=6a</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a^2\sdot\left(\frac{3}{4}\sdot a^2\right)=6a</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find one number]]
+
[[category: #find a number]]
 
[[comment: a²·¾a²=6a]]
 
[[comment: a²·¾a²=6a]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{2}{5}a^2=a</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: ⅖a²=a]]
 
}}</div></div>
 
}}</div></div>
 
<br>
 
<br>
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a+28=2a^2</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a+28=2a^2</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find one number]]
+
[[category: #find a number]]
 
[[comment: a+28=2a²]]
 
[[comment: a+28=2a²]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a^2=4\sdot\left(a+8\right)</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: a²=4(a+8)]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a^2=\left(3+\frac{10}{13}\right)\sdot\left(a+6\right)</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: a²=(3+¹⁰/₁₃)(a+6)]]
 
}}</div></div>
 
}}</div></div>
 
<br>
 
<br>
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a^2\sdot a=12a+10a^2</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a^2\sdot a=12a+10a^2</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find one number]]
+
[[category: #find a number]]
 
[[comment: a²·a=12a+10a²]]
 
[[comment: a²·a=12a+10a²]]
 
}}</div></div>
 
}}</div></div>
Line 75: Line 192:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sqrt{9a}=12</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sqrt{9a}=12</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find one number]]
+
[[category: #find a number]]
 
[[comment: √(9a)=12]]
 
[[comment: √(9a)=12]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sqrt{a}+\sqrt{a-\sqrt{a}}=3</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: √a+√(a-√a)=3]]
 
}}</div></div>
 
}}</div></div>
 
<br>
 
<br>
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(a\sdot\frac{2}{3}a\right)^2=\sqrt{50}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(a\sdot\frac{2}{3}a\right)^2=\sqrt{50}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find one number]]
+
[[category: #find a number]]
 
[[comment: (a·⅔a)²=√50]]
 
[[comment: (a·⅔a)²=√50]]
 
}}</div></div>
 
}}</div></div>
Line 87: Line 210:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(a\sdot\frac{2}{3}a\right)\sdot a=a\sdot\sqrt{8}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(a\sdot\frac{2}{3}a\right)\sdot a=a\sdot\sqrt{8}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find one number]]
+
[[category: #find a number]]
 
[[comment: (a·⅔a)·a=a√8]]
 
[[comment: (a·⅔a)·a=a√8]]
 
}}</div></div>
 
}}</div></div>
Line 95: Line 218:
 
[[category: #find a number]]
 
[[category: #find a number]]
 
[[comment: ³/₇a+⁴/₅a=⅔a+¼a+20]]
 
[[comment: ³/₇a+⁴/₅a=⅔a+¼a+20]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{1}{4}\sdot a=\frac{1}{3}\left[a-\left(\frac{1}{4}\sdot a\right)\right]=\frac{1}{2}\sdot\left[\left[a-\left(\frac{1}{4}\sdot a\right)\right]-\frac{1}{3}\left[a-\left(\frac{1}{4}\sdot a\right)\right]\right]</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: ¼a=⅓(a-¼a)=½((a-¼a)-⅓(a-¼a))]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left[\left[X-\left(\frac{1}{2}X+\frac{1}{2}\right)\right]-\left[\left[\frac{1}{2}\sdot\left[X-\left(\frac{1}{2}X+\frac{1}{2}\right)\right]\right]+\frac{1}{2}\right]\right]-\left[\frac{1}{2}\sdot\left[\left[X-\left(\frac{1}{2}X+\frac{1}{2}\right)\right]-\left[\left[\frac{1}{2}\sdot\left[X-\left(\frac{1}{2}X+\frac{1}{2}\right)\right]\right]+\frac{1}{2}\right]\right]+\frac{1}{2}\right]=1</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: a-(½a+½)-(½(a-(½a+½))+½)-(½(a-(½a+½)-(½(a-(½a+½))+½))+½)]]
 
}}</div></div>
 
}}</div></div>
 
<br>
 
<br>
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{3a}{4}:4=4:6</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{3a}{4}:4=4:6</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find one number]]
+
[[category: #find a number]]
 
[[comment: (3a/4)/4=4/6]]
 
[[comment: (3a/4)/4=4/6]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle2:3=4:a</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: 2/3=4/a]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle4=\frac{1}{3}\sdot15\longrightarrow a=\frac{1}{5}\sdot25</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: 4=⅓·15→a=⅕·25]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a^2=40+\frac{41}{b}</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: a²=40+41\b]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{4a}{5}=6+\frac{3}{5}</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: 4a/5=6⅗]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle4a\bmod5=1</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: 4a(mod)5=1]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle4a\bmod6=1</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: 4a(mod)6=1]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a\bmod3=a\bmod5=a\bmod7=1</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: a(mod)3=a(mod5)=a(mod7)=1]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a\bmod3=2;a\bmod5=4;a\bmod7=6</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: a(mod)3=2,a(mod5)=4,a(mod7)=6]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a\bmod2=1;a\bmod3=1;a\bmod4=1;a\bmod5=0</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: a(mod2)=1,a(mod3)=1,a(mod)4=1,a(mod5)=0]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a\bmod2=1;a\bmod3=2;a\bmod4=3;a\bmod5=0</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: a(mod2)=1,a(mod3)=2,a(mod)4=3,a(mod5)=0]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a\bmod2=1;a\bmod3=2;a\bmod4=3;a\bmod5=4;a\bmod6=5;a\bmod7=0</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: a(mod2)=1,a(mod3)=2,a(mod)4=3,a(mod5)=4,a(mod6)=5,a(mod7)=0]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a\bmod2=1;a\bmod3=1;a\bmod4=1;a\bmod5=1;a\bmod6=1;a\bmod7=0</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: a(mod2)=1,a(mod3)=1,a(mod)4=1,a(mod5)=1,a(mod6)=1,a(mod7)=0]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a\bmod1=0;a\bmod2=1;a\bmod3=2;a\bmod4=3;a\bmod5=4;a\bmod6=5;a\bmod7=1</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: a(mod)1=0,a(mod2)=1,a(mod3)=2,a(mod)4=3,a(mod5)=4,a(mod6)=5,a(mod7)=1]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a\bmod2=1;a\bmod3=2;a\bmod4=3;a\bmod5=1;a\bmod6=5;a\bmod7=1;a\bmod8=7;a\bmod9=8;a\bmod10=1</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: a(mod2)=1,a(mod3)=2,a(mod)4=3,a(mod5)=1,a(mod6)=5,a(mod7)=1,a(mod8)=7,a(mod9)=8,a(mod10)=1]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a\bmod2=0;a\bmod3=2;a\bmod4=0;a\bmod5=0;a\bmod6=2;a\bmod3=1;a\bmod8=4;a\bmod9=5;a\bmod10=0</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: a(mod2)=0,a(mod3)=2,a(mod)4=0,a(mod5)=0,a(mod6)=2,a(mod7)=3,a(mod8)=4,a(mod9)=5,a(mod10)=0]]
 +
}}</div></div>
 +
<br>
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a^2+19=b^2</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: a²+19=b²]]
 +
}}</div></div>
 +
<br>
 +
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a^2-15=b^2</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: a²-15=b²]]
 +
}}</div></div>
 +
<br>
 +
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a+b=n^2\\\scriptstyle a-b=m^2\end{cases}</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: a+b=n², a-b=m²]]
 +
}}</div></div>
 +
<br>
 +
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a+4=n^2\\\scriptstyle a-4=m^2\end{cases}</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: a+4=n², a-4=m²]]
 +
}}</div></div>
 +
<br>
 +
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a^2-5a=n^2\\\scriptstyle a^2+7a=m^2\end{cases}</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: a²-5a=n², a²+7a=m²]]
 
}}</div></div>
 
}}</div></div>
 
<br>
 
<br>
 +
 +
<div class="mw-collapsible mw-collapsed">repeated subtraction<div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: repeated subtraction]]
 +
}}</div></div>
  
 
== Find Two Numbers ==
 
== Find Two Numbers ==
Line 108: Line 373:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle b-1=a+1\\\scriptstyle2\sdot\left(a-1\right)=b+1\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle b-1=a+1\\\scriptstyle2\sdot\left(a-1\right)=b+1\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: b-1=a+1;2(a-1)=b+1]]
 
[[comment: b-1=a+1;2(a-1)=b+1]]
 
}}</div></div>
 
}}</div></div>
Line 114: Line 379:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle b-1=a+1\\\scriptstyle20\sdot\left(a-1\right)=b+1\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle b-1=a+1\\\scriptstyle20\sdot\left(a-1\right)=b+1\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: b-1=a+1;20(a-1)=b+1]]
 
[[comment: b-1=a+1;20(a-1)=b+1]]
 
}}</div></div>
 
}}</div></div>
Line 120: Line 385:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle2\sdot\left(b-1\right)=a+1\\\scriptstyle3\sdot\left(a-1\right)=b+1\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle2\sdot\left(b-1\right)=a+1\\\scriptstyle3\sdot\left(a-1\right)=b+1\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: 2(b-1)=a+1;3(a-1)=b+1]]
 
[[comment: 2(b-1)=a+1;3(a-1)=b+1]]
 
}}</div></div>
 
}}</div></div>
Line 126: Line 391:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a^2=b-5\\\scriptstyle a^2+b^2=100\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a^2=b-5\\\scriptstyle a^2+b^2=100\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a²=b-5;a²+b²=100]]
 
[[comment: a²=b-5;a²+b²=100]]
 
}}</div></div>
 
}}</div></div>
Line 132: Line 397:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a^2=2b+8\\\scriptstyle a^2+b^2=80\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a^2=2b+8\\\scriptstyle a^2+b^2=80\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a²=2b+8;a²+b²=80]]
 
[[comment: a²=2b+8;a²+b²=80]]
 
}}</div></div>
 
}}</div></div>
−
 
  
 
=== b=2a ===
 
=== b=2a ===
Line 141: Line 405:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle b=2a\\\scriptstyle a^2\sdot b=9b\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle b=2a\\\scriptstyle a^2\sdot b=9b\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: b=2a;a²b=9b]]
 
[[comment: b=2a;a²b=9b]]
 
}}</div></div>
 
}}</div></div>
Line 149: Line 413:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle4a+4b=\sqrt{100}\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle4a+4b=\sqrt{100}\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a/b=⅔;4a+4b=√100]]
 
[[comment: a/b=⅔;4a+4b=√100]]
 
}}</div></div>
 
}}</div></div>
Line 155: Line 419:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle a\sdot b=\sqrt{12}\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle a\sdot b=\sqrt{12}\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a/b=⅔;ab=√12]]
 
[[comment: a/b=⅔;ab=√12]]
 
}}</div></div>
 
}}</div></div>
Line 161: Line 425:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(3a+4b\right)\sdot\sqrt{5}=40\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(3a+4b\right)\sdot\sqrt{5}=40\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a/b=⅔;(3a+4b)√5=40]]
 
[[comment: a/b=⅔;(3a+4b)√5=40]]
 
}}</div></div>
 
}}</div></div>
Line 167: Line 431:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(a\sdot b\right)\sdot\sqrt{8}=100\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(a\sdot b\right)\sdot\sqrt{8}=100\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a/b=⅔;ab√8=100]]
 
[[comment: a/b=⅔;ab√8=100]]
 
}}</div></div>
 
}}</div></div>
Line 173: Line 437:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle a\sdot\sqrt{a}=2b\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle a\sdot\sqrt{a}=2b\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a/b=⅔;a√a=2b]]
 
[[comment: a/b=⅔;a√a=2b]]
 
}}</div></div>
 
}}</div></div>
Line 179: Line 443:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle a\sdot\sqrt{a}=b^2\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle a\sdot\sqrt{a}=b^2\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a/b=⅔;a√a=b²]]
 
[[comment: a/b=⅔;a√a=b²]]
 
}}</div></div>
 
}}</div></div>
Line 185: Line 449:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(a\sdot b\right)^2=b\sdot\sqrt{8}\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(a\sdot b\right)^2=b\sdot\sqrt{8}\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a/b=⅔;(ab)²=b√8]]
 
[[comment: a/b=⅔;(ab)²=b√8]]
 
}}</div></div>
 
}}</div></div>
Line 191: Line 455:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle a^2\sdot b=b\sdot\sqrt{b}\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle a^2\sdot b=b\sdot\sqrt{b}\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a/b=⅔;a²b=b√b]]
 
[[comment: a/b=⅔;a²b=b√b]]
 
}}</div></div>
 
}}</div></div>
Line 197: Line 461:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle5a+7b=16+\sqrt{8}\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle5a+7b=16+\sqrt{8}\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a/b=⅔;5a+7b=16+√8]]
 
[[comment: a/b=⅔;5a+7b=16+√8]]
 
}}</div></div>
 
}}</div></div>
Line 203: Line 467:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle3a+4\sqrt{b}=30\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle3a+4\sqrt{b}=30\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a/b=⅔;3a+4√b=30]]
 
[[comment: a/b=⅔;3a+4√b=30]]
 
}}</div></div>
 
}}</div></div>
Line 209: Line 473:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle a\sdot b=20+\sqrt{8}\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle a\sdot b=20+\sqrt{8}\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a/b=⅔;ab=20+√8]]
 
[[comment: a/b=⅔;ab=20+√8]]
 
}}</div></div>
 
}}</div></div>
Line 215: Line 479:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(a\sdot b\right)\sdot b=100+\sqrt{8}\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(a\sdot b\right)\sdot b=100+\sqrt{8}\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a/b=⅔;(ab)b=100+\sqrt{8}]]
 
[[comment: a/b=⅔;(ab)b=100+\sqrt{8}]]
 
}}</div></div>
 
}}</div></div>
Line 221: Line 485:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(a^2\sdot b\right)+\sqrt{a^2\sdot b}=342\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(a^2\sdot b\right)+\sqrt{a^2\sdot b}=342\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a/b=⅔;a²b+\sqrt{a²b}=342]]
 
[[comment: a/b=⅔;a²b+\sqrt{a²b}=342]]
 
}}</div></div>
 
}}</div></div>
Line 227: Line 491:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(a\sdot b\right)^2=20+\sqrt{30}\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(a\sdot b\right)^2=20+\sqrt{30}\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a/b=⅔;(ab)²=20+√30]]
 
[[comment: a/b=⅔;(ab)²=20+√30]]
 
}}</div></div>
 
}}</div></div>
Line 233: Line 497:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(a\sdot b\right)^2+\left(a^2\sdot\sqrt{4}\right)=4+\frac{1}{4}\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(a\sdot b\right)^2+\left(a^2\sdot\sqrt{4}\right)=4+\frac{1}{4}\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a/b=⅔;(ab)²+a²√4=4¼]]
 
[[comment: a/b=⅔;(ab)²+a²√4=4¼]]
 
}}</div></div>
 
}}</div></div>
Line 239: Line 503:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle3a+4b=\sqrt[3]{216}\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle3a+4b=\sqrt[3]{216}\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a/b=⅔;3a+4b=³√216]]
 
[[comment: a/b=⅔;3a+4b=³√216]]
 
}}</div></div>
 
}}</div></div>
Line 248: Line 512:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle a^2\sdot b=288\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle a^2\sdot b=288\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a/b=¾;a²b=288]]
 
[[comment: a/b=¾;a²b=288]]
 
}}</div></div>
 
}}</div></div>
Line 254: Line 518:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle\left(a\sdot b\right)\sdot\left(a+b\right)=a^2\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle\left(a\sdot b\right)\sdot\left(a+b\right)=a^2\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a/b=¾;(ab)·(a+b)=a²]]
 
[[comment: a/b=¾;(ab)·(a+b)=a²]]
 
}}</div></div>
 
}}</div></div>
Line 260: Line 524:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle a^2\sdot b=\sqrt{20+\frac{1}{4}}\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle a^2\sdot b=\sqrt{20+\frac{1}{4}}\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a/b=¾;a²b=√(20¼)]]
 
[[comment: a/b=¾;a²b=√(20¼)]]
 
}}</div></div>
 
}}</div></div>
Line 266: Line 530:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle8a=6\sqrt{b}\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle8a=6\sqrt{b}\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a/b=¾;8a=6√b]]
 
[[comment: a/b=¾;8a=6√b]]
 
}}</div></div>
 
}}</div></div>
Line 272: Line 536:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle\left(a\sdot\sqrt{8}\right)+b^2=48\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle\left(a\sdot\sqrt{8}\right)+b^2=48\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a/b=¾;a√8+b²=48]]
 
[[comment: a/b=¾;a√8+b²=48]]
 
}}</div></div>
 
}}</div></div>
Line 278: Line 542:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle\left(a\sdot b\right)^2+27=9b^2\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle\left(a\sdot b\right)^2+27=9b^2\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a/b=¾;(ab)²+27=9b²]]
 
[[comment: a/b=¾;(ab)²+27=9b²]]
 
}}</div></div>
 
}}</div></div>
Line 284: Line 548:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle\sqrt[3]{2a+3b}=8\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle\sqrt[3]{2a+3b}=8\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a/b=¾;³√(2a+3b)=8]]
 
[[comment: a/b=¾;³√(2a+3b)=8]]
 
}}</div></div>
 
}}</div></div>
Line 293: Line 557:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=4:5\\\scriptstyle\left(a\sdot b\right)^2=a^2\sdot b\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=4:5\\\scriptstyle\left(a\sdot b\right)^2=a^2\sdot b\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers[0] ]]
+
[[category: #find a number]]
 
[[comment: a/b=⅘;(ab)²=a²b]]
 
[[comment: a/b=⅘;(ab)²=a²b]]
 
}}</div></div>
 
}}</div></div>
Line 299: Line 563:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=4:5\\\scriptstyle a\sdot b=b\sdot\sqrt{8}\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=4:5\\\scriptstyle a\sdot b=b\sdot\sqrt{8}\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers[0] ]]
+
[[category: #find a number]]
 
[[comment: a/b=⅘;ab=b√8]]
 
[[comment: a/b=⅘;ab=b√8]]
 
}}</div></div>
 
}}</div></div>
Line 307: Line 571:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:5\\\scriptstyle\left(a\sdot b\right)^2=a^2\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:5\\\scriptstyle\left(a\sdot b\right)^2=a^2\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a/b=⅗;(ab)²=a²]]
 
[[comment: a/b=⅗;(ab)²=a²]]
 
}}</div></div>
 
}}</div></div>
Line 313: Line 577:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:5\\\scriptstyle\left[\left(a\sdot\sqrt{a}\right)+\left(b\sdot\sqrt{b}\right)\right]\sdot\sqrt{8}=100\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:5\\\scriptstyle\left[\left(a\sdot\sqrt{a}\right)+\left(b\sdot\sqrt{b}\right)\right]\sdot\sqrt{8}=100\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a/b=⅗;(a√a+b√b)√8=100]]
 
[[comment: a/b=⅗;(a√a+b√b)√8=100]]
 
}}</div></div>
 
}}</div></div>
Line 319: Line 583:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:5\\\scriptstyle\left(a\sdot b\right)\sdot\sqrt{8}=b\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:5\\\scriptstyle\left(a\sdot b\right)\sdot\sqrt{8}=b\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a/b=⅗;ab√8=b]]
 
[[comment: a/b=⅗;ab√8=b]]
 
}}</div></div>
 
}}</div></div>
Line 325: Line 589:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:5\\\scriptstyle a^2\sdot b=b^2\sdot\sqrt{8}\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:5\\\scriptstyle a^2\sdot b=b^2\sdot\sqrt{8}\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a/b=⅗;a²b=b²√8]]
 
[[comment: a/b=⅗;a²b=b²√8]]
 
}}</div></div>
 
}}</div></div>
Line 331: Line 595:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:5\\\scriptstyle a\sdot b=\sqrt[3]{729}\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:5\\\scriptstyle a\sdot b=\sqrt[3]{729}\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a/b=⅗;ab=³√729]]
 
[[comment: a/b=⅗;ab=³√729]]
 
}}</div></div>
 
}}</div></div>
Line 340: Line 604:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:5\\\scriptstyle a\sdot b=8\sqrt{b}\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:5\\\scriptstyle a\sdot b=8\sqrt{b}\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a/b=⅖;ab=8√b]]
 
[[comment: a/b=⅖;ab=8√b]]
 
}}</div></div>
 
}}</div></div>
Line 349: Line 613:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=1:4\\\scriptstyle\left(a\sdot b\right)\sdot a=\sqrt[3]{216}\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=1:4\\\scriptstyle\left(a\sdot b\right)\sdot a=\sqrt[3]{216}\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a/b=¼;aba=³√216]]
 
[[comment: a/b=¼;aba=³√216]]
 
}}</div></div>
 
}}</div></div>
Line 358: Line 622:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=1:3\\\scriptstyle\sqrt[3]{a}\sdot\sqrt[3]{b}=100\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=1:3\\\scriptstyle\sqrt[3]{a}\sdot\sqrt[3]{b}=100\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a/b=⅓;³√a³√b=100]]
 
[[comment: a/b=⅓;³√a³√b=100]]
 
}}</div></div>
 
}}</div></div>
Line 367: Line 631:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=5:7\\\scriptstyle\left(a\sdot b\right)^2=a^2\sdot\sqrt{8}\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=5:7\\\scriptstyle\left(a\sdot b\right)^2=a^2\sdot\sqrt{8}\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find two numbers]]
+
[[category: #find a number]]
 
[[comment: a/b=⁵/₇;(ab)²=a²√8]]
 
[[comment: a/b=⁵/₇;(ab)²=a²√8]]
 +
}}</div></div>
 +
<br>
 +
 +
=== 8:a:b:27 ===
 +
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle8:a:b:27</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: 8/a/b/27]]
 
}}</div></div>
 
}}</div></div>
 
<br>
 
<br>
Line 376: Line 649:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a+b=c\\\scriptstyle a+c=100b\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a+b=c\\\scriptstyle a+c=100b\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find three numbers]]
+
[[category: #find a number]]
 
[[comment: a+b=c;a+c=100b]]
 
[[comment: a+b=c;a+c=100b]]
 
}}</div></div><br>
 
}}</div></div><br>
Line 382: Line 655:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle b:c=3:4\\\scriptstyle\left(a^2\sdot a\right)+b^2=\left(12+\frac{1}{2}\right)\sdot c\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle b:c=3:4\\\scriptstyle\left(a^2\sdot a\right)+b^2=\left(12+\frac{1}{2}\right)\sdot c\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #a/b/c;2/3/4]]
+
[[category: #find a number]]
 
[[comment: (a²·a)+b²=12½c]]
 
[[comment: (a²·a)+b²=12½c]]
 
}}</div></div><br>
 
}}</div></div><br>
Line 388: Line 661:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle\left(a+b\right):c=3+\frac{2}{5}+\frac{1}{7}\\\scriptstyle\left(a+c\right):b=7+\frac{2}{3}+\frac{1}{4}\\\scriptstyle b=30\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle\left(a+b\right):c=3+\frac{2}{5}+\frac{1}{7}\\\scriptstyle\left(a+c\right):b=7+\frac{2}{3}+\frac{1}{4}\\\scriptstyle b=30\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find three numbers]]
+
[[category: #find a number]]
 
[[comment: (a+b)\c=3⅖⅐;(a+c)\b=7⅔¼;b=30]]
 
[[comment: (a+b)\c=3⅖⅐;(a+c)\b=7⅔¼;b=30]]
 
}}</div></div><br>
 
}}</div></div><br>
Line 394: Line 667:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle\left(a+b\right):c=\frac{3}{5}+\frac{1}{6}\\\scriptstyle\left(a+c\right):b=2+\frac{1}{3}\\\scriptstyle a=20\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle\left(a+b\right):c=\frac{3}{5}+\frac{1}{6}\\\scriptstyle\left(a+c\right):b=2+\frac{1}{3}\\\scriptstyle a=20\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find three numbers]]
+
[[category: #find a number]]
 
[[comment: (a+b)/c=⅗+⅙;(a+c)/b=2⅓;a=20]]
 
[[comment: (a+b)/c=⅗+⅙;(a+c)/b=2⅓;a=20]]
 
}}</div></div><br>
 
}}</div></div><br>
Line 400: Line 673:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle b:\left(c-a\right)=3+\frac{1}{3}\\\scriptstyle c:\left(b-a\right)=6\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle b:\left(c-a\right)=3+\frac{1}{3}\\\scriptstyle c:\left(b-a\right)=6\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find three numbers]]
+
[[category: #find a number]]
 
[[comment: b/(c-a)=3⅓;c/(b-a)=6]]
 
[[comment: b/(c-a)=3⅓;c/(b-a)=6]]
 
}}</div></div><br>
 
}}</div></div><br>
Line 406: Line 679:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle\left(a+b\right):\left(c-a\right)=4+\frac{1}{2}\\\scriptstyle\left(a+c\right):\left(b-a\right)=5\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle\left(a+b\right):\left(c-a\right)=4+\frac{1}{2}\\\scriptstyle\left(a+c\right):\left(b-a\right)=5\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find three numbers]]
+
[[category: #find a number]]
 
[[comment: (a+b)/(c-a)=4½;(a+c)/(b-a)=5]]
 
[[comment: (a+b)/(c-a)=4½;(a+c)/(b-a)=5]]
 
}}</div></div><br>
 
}}</div></div><br>
Line 412: Line 685:
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a+\frac{1}{4}\sdot\left(b+c\right)=b+\frac{1}{6}\sdot\left(a+c\right)=c+\frac{1}{9}\sdot\left(a+b\right)\\\scriptstyle b=20\end{cases}</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a+\frac{1}{4}\sdot\left(b+c\right)=b+\frac{1}{6}\sdot\left(a+c\right)=c+\frac{1}{9}\sdot\left(a+b\right)\\\scriptstyle b=20\end{cases}</math><div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #find three numbers]]
+
[[category: #find a number]]
 
[[comment: a+¼(b+c)=b+⅙(a+c)=c+⅑(a+b);b=20]]
 
[[comment: a+¼(b+c)=b+⅙(a+c)=c+⅑(a+b);b=20]]
 
}}</div></div><br>
 
}}</div></div><br>
  
−
<div class="mw-collapsible mw-collapsed">three proportional numbers, a·b·c=1000<div class="mw-collapsible-content">
+
== Three Proportional Numbers==
 +
 
 +
<div class="mw-collapsible mw-collapsed">three proportional numbers<div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: three proportional numbers]]
 +
}}</div></div><br>
 +
 
 +
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=b:c\\\scriptstyle a\sdot b\sdot c=1000\end{cases}</math><div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: a·b·c=1000;a\b\c]]
 +
}}</div></div><br>
 +
 
 +
== Four Proportional Numbers==
 +
 
 +
<div class="mw-collapsible mw-collapsed">four proportional numbers<div class="mw-collapsible-content">
 
{{#annotask:
 
{{#annotask:
−
[[category: #three proportional numbers]]
+
[[category: #find a number]]
−
[[comment: a·b·c=1000]]
+
[[comment: four proportional numbers]]
 +
}}</div></div><br>
 +
 
 +
== Five Proportional Numbers ==
 +
 
 +
<div class="mw-collapsible mw-collapsed">five proportional numbers<div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: five proportional numbers]]
 +
}}</div></div><br>
 +
 
 +
== Six Proportional Numbers ==
 +
 
 +
<div class="mw-collapsible mw-collapsed">six proportional numbers<div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: six proportional numbers]]
 +
}}</div></div><br>
 +
 
 +
== Seven Proportional Numbers ==
 +
 
 +
<div class="mw-collapsible mw-collapsed">seven proportional numbers<div class="mw-collapsible-content">
 +
{{#annotask:
 +
[[category: #find a number]]
 +
[[comment: seven proportional numbers]]
 
}}</div></div><br>
 
}}</div></div><br>

Latest revision as of 14:40, 12 July 2024

Find a Perfect Number

[Expand]perfect number

Find One Number

[Expand]\scriptstyle37a=49


[Expand]\scriptstyle9a=1


[Expand]\scriptstyle13\sdot7\sdot3\sdot a=3


[Expand]\scriptstyle\frac{7a}{9}=49


[Expand]\scriptstyle\frac{a}{5}\sdot8=23


[Expand]\scriptstyle\frac{\frac{1}{5}a}{\sqrt{a}}=7


[Expand]\scriptstyle\frac{3}{a}=5+\frac{7}{b}


[Expand]\scriptstyle\left(\frac{1}{3}\sdot a\right)+\left(\frac{1}{5}\sdot a\right)=10


[Expand]\scriptstyle\left(\frac{1}{3}\sdot a\right)+\left(\frac{1}{4}\sdot a\right)=10+\frac{1}{2}


[Expand]\scriptstyle\left(\frac{1}{3}\sdot a\right)+\left(\frac{1}{4}\sdot a\right)=20


[Expand]\scriptstyle\left(\frac{1}{3}\sdot a\right)+\left(\frac{1}{4}\sdot a\right)=6


[Expand]\scriptstyle\left(\frac{1}{4}\sdot a\right)+\left(\frac{1}{5}\sdot a\right)=5


[Expand]\scriptstyle a+\left(\frac{1}{3}\sdot a\right)+\left(\frac{2}{5}\sdot a\right)=81


[Expand]\scriptstyle a+\left[\left(2+\frac{2}{3}\right)\sdot a\right]=11


[Expand]\scriptstyle\left(\frac{1}{2}\sdot a\right)+\left(\frac{1}{3}\sdot a\right)+\left(\frac{1}{4}\sdot a\right)-5=24


[Expand]\scriptstyle a+\frac{1}{2}\sdot\left(\frac{1}{4}\sdot a\right)+\left(\frac{3}{4}\sdot a\right)+\frac{1}{4}\sdot\left(\frac{1}{4}\sdot a\right)=2


[Expand]\scriptstyle a+\frac{1}{2}a+\frac{1}{5}a+\frac{1}{7}a=2


[Expand]\scriptstyle a+\frac{1}{2}\sdot\frac{1}{2}a+\frac{1}{3}\sdot\frac{1}{3}a+\frac{1}{5}\sdot\frac{1}{5}a=7


[Expand]\scriptstyle\left(a+\frac{1}{3}a\right)+\left[\frac{2}{5}\sdot\left(a+\frac{1}{3}a\right)\right]+\left[\frac{3}{4}\sdot\left(a+\frac{1}{3}a\right)+\left[\frac{2}{5}\sdot\left(a+\frac{1}{3}a\right)\right]\right]=13


[Expand]\scriptstyle a-\left(\frac{1}{3}\sdot a\right)-\left(\frac{1}{4}\sdot a\right)=29


[Expand]\scriptstyle\left(\frac{1}{2}a+\frac{1}{3}a\right)-\left(\frac{1}{4}a+\frac{1}{5}a\right)=4


[Expand]\scriptstyle a-\left(\frac{1}{3}\sdot a\right)=\sqrt{a}


[Expand]\scriptstyle\left[a-\left(\frac{2}{3}\sdot a\right)\right]^2=20


[Expand]\scriptstyle\left[a\sdot\left(\frac{2}{3}\sdot a\right)\right]^2=36


[Expand]\scriptstyle a^2\sdot\left(\frac{3}{4}\sdot a^2\right)=6a


[Expand]\scriptstyle\frac{2}{5}a^2=a


[Expand]\scriptstyle a+28=2a^2


[Expand]\scriptstyle a^2=4\sdot\left(a+8\right)


[Expand]\scriptstyle a^2=\left(3+\frac{10}{13}\right)\sdot\left(a+6\right)


[Expand]\scriptstyle a^2\sdot a=12a+10a^2


[Expand]\scriptstyle\sqrt{9a}=12


[Expand]\scriptstyle\sqrt{a}+\sqrt{a-\sqrt{a}}=3


[Expand]\scriptstyle\left(a\sdot\frac{2}{3}a\right)^2=\sqrt{50}


[Expand]\scriptstyle\left(a\sdot\frac{2}{3}a\right)\sdot a=a\sdot\sqrt{8}


[Expand]\scriptstyle\frac{3}{7}a+\frac{4}{5}a=\frac{2}{3}a+\frac{1}{4}a+20


[Expand]\scriptstyle\frac{1}{4}\sdot a=\frac{1}{3}\left[a-\left(\frac{1}{4}\sdot a\right)\right]=\frac{1}{2}\sdot\left[\left[a-\left(\frac{1}{4}\sdot a\right)\right]-\frac{1}{3}\left[a-\left(\frac{1}{4}\sdot a\right)\right]\right]


[Expand]\scriptstyle\left[\left[X-\left(\frac{1}{2}X+\frac{1}{2}\right)\right]-\left[\left[\frac{1}{2}\sdot\left[X-\left(\frac{1}{2}X+\frac{1}{2}\right)\right]\right]+\frac{1}{2}\right]\right]-\left[\frac{1}{2}\sdot\left[\left[X-\left(\frac{1}{2}X+\frac{1}{2}\right)\right]-\left[\left[\frac{1}{2}\sdot\left[X-\left(\frac{1}{2}X+\frac{1}{2}\right)\right]\right]+\frac{1}{2}\right]\right]+\frac{1}{2}\right]=1


[Expand]\scriptstyle\frac{3a}{4}:4=4:6


[Expand]\scriptstyle2:3=4:a


[Expand]\scriptstyle4=\frac{1}{3}\sdot15\longrightarrow a=\frac{1}{5}\sdot25


[Expand]\scriptstyle a^2=40+\frac{41}{b}


[Expand]\scriptstyle\frac{4a}{5}=6+\frac{3}{5}


[Expand]\scriptstyle4a\bmod5=1


[Expand]\scriptstyle4a\bmod6=1


[Expand]\scriptstyle a\bmod3=a\bmod5=a\bmod7=1


[Expand]\scriptstyle a\bmod3=2;a\bmod5=4;a\bmod7=6


[Expand]\scriptstyle a\bmod2=1;a\bmod3=1;a\bmod4=1;a\bmod5=0


[Expand]\scriptstyle a\bmod2=1;a\bmod3=2;a\bmod4=3;a\bmod5=0


[Expand]\scriptstyle a\bmod2=1;a\bmod3=2;a\bmod4=3;a\bmod5=4;a\bmod6=5;a\bmod7=0


[Expand]\scriptstyle a\bmod2=1;a\bmod3=1;a\bmod4=1;a\bmod5=1;a\bmod6=1;a\bmod7=0


[Expand]\scriptstyle a\bmod1=0;a\bmod2=1;a\bmod3=2;a\bmod4=3;a\bmod5=4;a\bmod6=5;a\bmod7=1


[Expand]\scriptstyle a\bmod2=1;a\bmod3=2;a\bmod4=3;a\bmod5=1;a\bmod6=5;a\bmod7=1;a\bmod8=7;a\bmod9=8;a\bmod10=1


[Expand]\scriptstyle a\bmod2=0;a\bmod3=2;a\bmod4=0;a\bmod5=0;a\bmod6=2;a\bmod3=1;a\bmod8=4;a\bmod9=5;a\bmod10=0


[Expand]\scriptstyle a^2+19=b^2


[Expand]\scriptstyle a^2-15=b^2


[Expand]\scriptstyle\begin{cases}\scriptstyle a+b=n^2\\\scriptstyle a-b=m^2\end{cases}


[Expand]\scriptstyle\begin{cases}\scriptstyle a+4=n^2\\\scriptstyle a-4=m^2\end{cases}


[Expand]\scriptstyle\begin{cases}\scriptstyle a^2-5a=n^2\\\scriptstyle a^2+7a=m^2\end{cases}


[Expand]repeated subtraction

Find Two Numbers

[Expand]\scriptstyle\begin{cases}\scriptstyle b-1=a+1\\\scriptstyle2\sdot\left(a-1\right)=b+1\end{cases}


[Expand]\scriptstyle\begin{cases}\scriptstyle b-1=a+1\\\scriptstyle20\sdot\left(a-1\right)=b+1\end{cases}


[Expand]\scriptstyle\begin{cases}\scriptstyle2\sdot\left(b-1\right)=a+1\\\scriptstyle3\sdot\left(a-1\right)=b+1\end{cases}


[Expand]\scriptstyle\begin{cases}\scriptstyle a^2=b-5\\\scriptstyle a^2+b^2=100\end{cases}


[Expand]\scriptstyle\begin{cases}\scriptstyle a^2=2b+8\\\scriptstyle a^2+b^2=80\end{cases}

b=2a

[Expand]\scriptstyle\begin{cases}\scriptstyle b=2a\\\scriptstyle a^2\sdot b=9b\end{cases}

a:b=2:3

[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle4a+4b=\sqrt{100}\end{cases}


[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle a\sdot b=\sqrt{12}\end{cases}


[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(3a+4b\right)\sdot\sqrt{5}=40\end{cases}


[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(a\sdot b\right)\sdot\sqrt{8}=100\end{cases}


[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle a\sdot\sqrt{a}=2b\end{cases}


[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle a\sdot\sqrt{a}=b^2\end{cases}


[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(a\sdot b\right)^2=b\sdot\sqrt{8}\end{cases}


[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle a^2\sdot b=b\sdot\sqrt{b}\end{cases}


[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle5a+7b=16+\sqrt{8}\end{cases}


[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle3a+4\sqrt{b}=30\end{cases}


[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle a\sdot b=20+\sqrt{8}\end{cases}


[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(a\sdot b\right)\sdot b=100+\sqrt{8}\end{cases}


[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(a^2\sdot b\right)+\sqrt{a^2\sdot b}=342\end{cases}


[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(a\sdot b\right)^2=20+\sqrt{30}\end{cases}


[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(a\sdot b\right)^2+\left(a^2\sdot\sqrt{4}\right)=4+\frac{1}{4}\end{cases}


[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle3a+4b=\sqrt[3]{216}\end{cases}


a:b=3:4

[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle a^2\sdot b=288\end{cases}


[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle\left(a\sdot b\right)\sdot\left(a+b\right)=a^2\end{cases}


[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle a^2\sdot b=\sqrt{20+\frac{1}{4}}\end{cases}


[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle8a=6\sqrt{b}\end{cases}


[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle\left(a\sdot\sqrt{8}\right)+b^2=48\end{cases}


[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle\left(a\sdot b\right)^2+27=9b^2\end{cases}


[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle\sqrt[3]{2a+3b}=8\end{cases}


a:b=4:5

[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=4:5\\\scriptstyle\left(a\sdot b\right)^2=a^2\sdot b\end{cases}


[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=4:5\\\scriptstyle a\sdot b=b\sdot\sqrt{8}\end{cases}

a:b=3:5

[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=3:5\\\scriptstyle\left(a\sdot b\right)^2=a^2\end{cases}


[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=3:5\\\scriptstyle\left[\left(a\sdot\sqrt{a}\right)+\left(b\sdot\sqrt{b}\right)\right]\sdot\sqrt{8}=100\end{cases}


[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=3:5\\\scriptstyle\left(a\sdot b\right)\sdot\sqrt{8}=b\end{cases}


[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=3:5\\\scriptstyle a^2\sdot b=b^2\sdot\sqrt{8}\end{cases}


[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=3:5\\\scriptstyle a\sdot b=\sqrt[3]{729}\end{cases}


a:b=2:5

[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=2:5\\\scriptstyle a\sdot b=8\sqrt{b}\end{cases}


a:b=1:4

[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=1:4\\\scriptstyle\left(a\sdot b\right)\sdot a=\sqrt[3]{216}\end{cases}


a:b=1:3

[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=1:3\\\scriptstyle\sqrt[3]{a}\sdot\sqrt[3]{b}=100\end{cases}


a:b=5:7

[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=5:7\\\scriptstyle\left(a\sdot b\right)^2=a^2\sdot\sqrt{8}\end{cases}


8:a:b:27

[Expand]\scriptstyle8:a:b:27


Find Three Numbers

[Expand]\scriptstyle\begin{cases}\scriptstyle a+b=c\\\scriptstyle a+c=100b\end{cases}

[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle b:c=3:4\\\scriptstyle\left(a^2\sdot a\right)+b^2=\left(12+\frac{1}{2}\right)\sdot c\end{cases}

[Expand]\scriptstyle\begin{cases}\scriptstyle\left(a+b\right):c=3+\frac{2}{5}+\frac{1}{7}\\\scriptstyle\left(a+c\right):b=7+\frac{2}{3}+\frac{1}{4}\\\scriptstyle b=30\end{cases}

[Expand]\scriptstyle\begin{cases}\scriptstyle\left(a+b\right):c=\frac{3}{5}+\frac{1}{6}\\\scriptstyle\left(a+c\right):b=2+\frac{1}{3}\\\scriptstyle a=20\end{cases}

[Expand]\scriptstyle\begin{cases}\scriptstyle b:\left(c-a\right)=3+\frac{1}{3}\\\scriptstyle c:\left(b-a\right)=6\end{cases}

[Expand]\scriptstyle\begin{cases}\scriptstyle\left(a+b\right):\left(c-a\right)=4+\frac{1}{2}\\\scriptstyle\left(a+c\right):\left(b-a\right)=5\end{cases}

[Expand]\scriptstyle\begin{cases}\scriptstyle a+\frac{1}{4}\sdot\left(b+c\right)=b+\frac{1}{6}\sdot\left(a+c\right)=c+\frac{1}{9}\sdot\left(a+b\right)\\\scriptstyle b=20\end{cases}

Three Proportional Numbers

[Expand]three proportional numbers

[Expand]\scriptstyle\begin{cases}\scriptstyle a:b=b:c\\\scriptstyle a\sdot b\sdot c=1000\end{cases}

Four Proportional Numbers

[Expand]four proportional numbers

Five Proportional Numbers

[Expand]five proportional numbers

Six Proportional Numbers

[Expand]six proportional numbers

Seven Proportional Numbers

[Expand]seven proportional numbers

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