Find Pairs

Since $15120 = 2^{4} \times 3^{3} \times 5 \times 7$ , to have positive integers $x,y$ such that $xy = 15120$ and gcf $(x,y) = 6$ we will require that $x = 6m$ and $y = 6n$ for positive integers $m,n$ such that $mn = 420 = 2^{2} \times 3 \times 5 \times 7$ and gcf $(m,n) = 1$ . To ensure that $m,n$ are relatively prime, we must assign each of $2^{2}, 3, 5$ and $7$ to the factorizations of one or the other of $m,n$ , (and thus in turn to $x,y$ ). This creates a total of $2^{4} = 16$ ordered pairs that satisfy the given conditions, but since we are asked for the number of unordered pairs we need to divide by $2$ to end up with an answer of $\boxed{8}$ .

The $8$ unordered pairs are $(6,2520), (18,840), (24,630), (30,504), (42,360), (72,210), (90,168)$ and $(120,126)$ .