2005 AMC 12B Problems/Problem 21
Problem
A positive integer has
divisors and
has
divisors. What is the greatest integer
such that
divides
?
Solution
If has
factors, then
is a product of
powers of (not necessarily distinct) primes. When multiplied by
, the amount of factors of
increased by
, so there are
possible powers of
in the factorization of
, and
possible powers of
in the factorization of
, which would be
,
, and
. Therefore the highest power of
that could divide
is
.