2005 AMC 10A Problems/Problem 24
Problem
For each positive integer , let
denote the greatest prime factor of
. For how many positive integers
is it true that both
and
?
Solution
If , then
, where
is a prime number.
If , then
, where
is a different prime number.
So:
Since :
.
Looking at pairs of factors of , we have several possibilities to solve for
and
:
The only solution where both numbers are primes is
.
Therefore the number of positive integers that satisfy both statements is