Question

Two numbers have sum `-37` and product `300`. What are the numbers?

Answer 1

`-12` and `-25`

Explanation:

The prime factorisation of `300` is:

`300 = 2^2*3*5^2`

If `m*n = 300` and `m+n = 37` then note that `37` is divisible by neither `2` nor `5`. So when splitting the factors of `300` into `m` and `n`, one of them must be divisible by `2^2 = 4` and one (possibly the same one) must be divisible by `5^2 = 25`.

We then see that `300/25 = 12` and `25+12 = 37`.

So the numbers we are looking for are `-12` and `-25`