Here is the problem. "How many ordered pairs $(x,y)$ of positive integers satisfy the equation $xy = 144$?"
I would consider this to be a fairly easy problem as you just have to find the number of factors. I did the prime factorization of $144$ and got $2^4 \cdot{3^2}$. With that, the number of factors is $(4 + 1) x (2 + 1) = 15$. Since it's odd, that means $144$ is a perfect square and therefore involves $7$ pairs of factors and another pair to account for the perfect square. So I got $8$ as my answer.
However the answer that is given is $15$. I'm not sure where I went wrong here.
Because $x$ and $y$ are separate variables, the ordered pairs $(1,144)$ and $(144,1)$ are different solutions.
So, there are $15$ solutions.