Answering my own question, much later: in its factored form, the expression is really a product of 5 consecutive integers. That means that it is guaranteed to have a factor of 5, a factor of 4, a factor of 3, and a factor of 2.

5 x 4 x 3 x 2 = 120.

Not sure anyone else was wondering what the answer to this was, but just in case.