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

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(Work in progress of my answer to this question.) |
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<math>\textbf{(A)} \text{ 0} \qquad \textbf{(B)} \text{ 2} \qquad \textbf{(C)} \text{ 4} \qquad \textbf{(D)} \text{ 6} \qquad \textbf{(E)} \text{ 8}</math> | <math>\textbf{(A)} \text{ 0} \qquad \textbf{(B)} \text{ 2} \qquad \textbf{(C)} \text{ 4} \qquad \textbf{(D)} \text{ 6} \qquad \textbf{(E)} \text{ 8}</math> | ||

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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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+ | <cmath>x\cdot y + 63 = 20x + 12y</cmath>, | ||

+ | <cmath>x\cdot y - 20x - 12y + 63 = 0</cmath>. | ||

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+ | (awesomeag) |

## Revision as of 15:22, 16 February 2018

23. How many ordered pairs of positive integers satisfy the equation where denotes the greatest common divisor of and , and denotes their least common multiple?

Let , and . Therefore, . Thus, the equation becomes

, .

(awesomeag)