If a and b are positive integers such that the greatest common factor of $a^2b^2$ and $ab^3$ is 45, then which of the following could b equal?
Select all such integers.
A. 3
B. 5
C. 9
D. 15
E. 45
I think the answer should be A C D E. I went through all 5 choices and test the possibilities. It does not seem to be very efficient during testing.
Note that $b$ cannot be any of the last four answers, else the gcd would not be $45$.
For example, if $b=5$, then the gcd is divisible by $25$. If $b=9$, the gcd is divisible by $81$.
Only $3$ is possible, with $a=5$.
Perhaps a little inefficient, but the same idea rules out the last four options.