2018 AMC 10B Problems/Problem 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?
Solution
Let $x = \lcm(a, b)$ (Error compiling LaTeX. ), and . Therefore, $a\cdot b = \lcm(a, b)\cdot gcd(a, b) = x\cdot y$ (Error compiling LaTeX. ). Thus, the equation becomes
Using Simon's Favorite Factoring Trick, we rewrite this equation as
Since and , we have and , or and . This gives us the solutions and . Obviously, the first pair does not work. Assume . We must have and , and we could then have , so there are solutions. (awesomeag)
Edited by IronicNinja~
See Also
2018 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 22 |
Followed by Problem 24 | |
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All AMC 10 Problems and Solutions |
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