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 , and
. Therefore,
. 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)
See Also
2018 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 22 |
Followed by Problem 24 | |
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