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Difference between revisions of "2018 AMC 10B Problems/Problem 23"

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==Problem==
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How many ordered pairs <math>(a, b)</math> of positive integers satisfy the equation  
 
How many ordered pairs <math>(a, b)</math> of positive integers satisfy the equation  
 
<cmath>a\cdot b + 63 = 20\cdot \text{lcm}(a, b) + 12\cdot\text{gcd}(a,b),</cmath>
 
<cmath>a\cdot b + 63 = 20\cdot \text{lcm}(a, b) + 12\cdot\text{gcd}(a,b),</cmath>
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==Solution==
 
==Solution==
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Let <math>x = lcm(a, b)</math>, and <math>y = gcd(a, b)</math>. Therefore, <math>a\cdot b = lcm(a, b)\cdot gcd(a, b) = x\cdot y</math>. Thus, the equation becomes
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Let <math>x = \text{lcm}(a, b)</math>, and <math>y = \text{gcd}(a, b)</math>. Therefore, <math>a\cdot b = \text{lcm}(a, b)\cdot \text{gcd}(a, b) = x\cdot y</math>. Thus, the equation becomes
  
 
<cmath>x\cdot y + 63 = 20x + 12y</cmath>
 
<cmath>x\cdot y + 63 = 20x + 12y</cmath>
 
<cmath>x\cdot y - 20x - 12y + 63 = 0</cmath>
 
<cmath>x\cdot y - 20x - 12y + 63 = 0</cmath>
  
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Using Simon's Favorite Factoring Trick, we rewrite this equation as
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Using [[Simon's Favorite Factoring Trick]], we rewrite this equation as
  
 
<cmath>(x - 12)(y - 20) - 240 + 63 = 0</cmath>
 
<cmath>(x - 12)(y - 20) - 240 + 63 = 0</cmath>
 
<cmath>(x - 12)(y - 20) = 177</cmath>
 
<cmath>(x - 12)(y - 20) = 177</cmath>
  
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From here we can already see that this is a quadratic, and thus must have 2 solutions. But, let's continue, to see if one of the solutions is extraneous.
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Since <math>177 = 3\cdot 59</math> and <math>x > y</math>, we have <math>x  - 12 = 59</math> and <math>y - 20 = 3</math>, or <math>x - 12 = 177</math> and <math>y - 20 = 1</math>. This gives us the solutions <math>(71, 23)</math> and <math>(189, 21)</math>. Since the <math>\text{GCD}</math> must be a divisor of the <math>\text{LCM}</math>, the first pair does not work. Assume <math>a>b</math>. We must have <math>a = 21 \cdot 9</math> and <math>b = 21</math>, and we could then have <math>a<b</math>, so there are <math>\boxed{\textbf{(B)} ~2}</math> solutions.
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(awesomeag)
  
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Since <math>177 = 3\cdot 59</math> and <math>x > y</math>, we have <math>x  - 12 = 59</math> and <math>y - 20 = 3</math>, or <math>x - 12 = 177</math> and <math>y - 20 = 1</math>. This gives us the solutions <math>(71, 23)</math> and <math>(189, 21)</math>. Since the GCD must be a divisor of the LCM, the first pair does not work. Assume <math>a>b</math>. We must have <math>a = 21 \cdot 9</math> and <math>b = 21</math>, and we could then have <math>a<b</math>, so there are <math>\boxed{2}</math> solutions.
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Edited by IronicNinja, Firebolt360, and mprincess0229~
−
(awesomeag)
+
 
 +
== Video Solution by OmegaLearn==
 +
https://youtu.be/zfChnbMGLVQ?t=494
  
−
Edited by IronicNinja and Firebolt360~
+
~ pi_is_3.14
  
 
==Video Solution==
 
==Video Solution==
 +
 
https://www.youtube.com/watch?v=JWGHYUeOx-k
 
https://www.youtube.com/watch?v=JWGHYUeOx-k
  

Latest revision as of 10:40, 14 October 2024

Problem

How many ordered pairs $(a, b)$ of positive integers satisfy the equation \[a\cdot b + 63 = 20\cdot \text{lcm}(a, b) + 12\cdot\text{gcd}(a,b),\] where $\text{gcd}(a,b)$ denotes the greatest common divisor of $a$ and $b$, and $\text{lcm}(a,b)$ denotes their least common multiple?

$\textbf{(A)} \text{ 0} \qquad \textbf{(B)} \text{ 2} \qquad \textbf{(C)} \text{ 4} \qquad \textbf{(D)} \text{ 6} \qquad \textbf{(E)} \text{ 8}$


Solution

Let $x = \text{lcm}(a, b)$, and $y = \text{gcd}(a, b)$. Therefore, $a\cdot b = \text{lcm}(a, b)\cdot \text{gcd}(a, b) = x\cdot y$. Thus, the equation becomes

\[x\cdot y + 63 = 20x + 12y\] \[x\cdot y - 20x - 12y + 63 = 0\]

Using Simon's Favorite Factoring Trick, we rewrite this equation as

\[(x - 12)(y - 20) - 240 + 63 = 0\] \[(x - 12)(y - 20) = 177\]

Since $177 = 3\cdot 59$ and $x > y$, we have $x - 12 = 59$ and $y - 20 = 3$, or $x - 12 = 177$ and $y - 20 = 1$. This gives us the solutions $(71, 23)$ and $(189, 21)$. Since the $\text{GCD}$ must be a divisor of the $\text{LCM}$, the first pair does not work. Assume $a>b$. We must have $a = 21 \cdot 9$ and $b = 21$, and we could then have $a<b$, so there are $\boxed{\textbf{(B)} ~2}$ solutions. (awesomeag)

Edited by IronicNinja, Firebolt360, and mprincess0229~

Video Solution by OmegaLearn

https://youtu.be/zfChnbMGLVQ?t=494

~ pi_is_3.14

Video Solution

https://www.youtube.com/watch?v=JWGHYUeOx-k

See Also

2018 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 22 		Followed by
Problem 24
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

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