NumTheSL5#16

The number 12 can be broken up to $x^2*y^2$, since the number of factors is $(2+1)(2+1)=3\cdot3=9.$ The three numbers do not work here since there are no possible outcomes.

Thus, we can have: $2^2*3^2=36.$

$2^2*5^2=100.$

$2^2*7^2=196.$

I do not think any more possible outcomes are there, and you can still check my work.

Thus, the answer is $36+100+196=\boxed{332}.$