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

Dec 8, 2016

$- 12$ and $- 25$

#### Explanation:

The prime factorisation of $300$ is:

$300 = {2}^{2} \cdot 3 \cdot {5}^{2}$

If $m \cdot n = 300$ 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 $\frac{300}{25} = 12$ and $25 + 12 = 37$.

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