Difference between revisions of "2018 AMC 10B Problems/Problem 23"
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+ | ==Video Solution== | ||
+ | https://www.youtube.com/watch?v=JWGHYUeOx-k | ||
==See Also== | ==See Also== | ||
{{AMC10 box|year=2018|ab=B|num-b=22|num-a=24}} | {{AMC10 box|year=2018|ab=B|num-b=22|num-a=24}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 03:29, 11 February 2019
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 lcm
, and
gcd
. Therefore,
lcm
gcd
. 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
. Since the Greatest Common Denominator must be a divisor of the Lowest Common Multiple, the first pair does not work. Assume
. We must have
and
, and we could then have
, so there are
solutions.
(awesomeag)
Edited by IronicNinja~
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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