Having some trouble wrapping my head around this one:
find the number of solutions to the equation $(x_1)(x_2)(x_3)(x_4) = 2016$, where $(x_i)$s are integers that are not necessarily positive.
Here's what I've been thinking so far:
Let's bring $2016$ down to its prime factorized form, such that we have $2016 = 2^5\times3^2\times7^1$.
Because the solution may include negative integers, note that a pair or 4 integers will form positive numbers, therefore we have $C(4,2)$ OR $C(4,4)$ ways of doing this.
Let's also note that the solution may include 1s in $3, 2, 1$, or $0$ positions for $(x_i)$, such that we have $C(4,3), C(4,2), C(4,1)$, or $C(4,0)$.
Now I'm thinking that we should look at each particular element of the prime factorization, such that we have elements $2, 2, 2, 2, 3, 3, 3, 7$ that are not entirely distinct. We should now think of the number of ways we can fill in $8$ elements into each of these $1, 2, 3$, or $4$ positions depending on the number of 1s we have.
And this is where I am having a little bit of trouble thinking through the solution. I'm thinking I could check the number of distinct integers that I could form with $2^5\times3^2\times7^1$ and then go from there? But I'm having trouble thinking through this methodology as well.