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

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

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Answer 1 · 2016-12-08T14:42:33

$- 12$ $-12$ and $- 25$ $-25$

Explanation:

The prime factorisation of $300$ $300$ is:

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

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

We then see that $\frac{300}{25} = 12$ $300/25 = 12$ and $25 + 12 = 37$ $25+12 = 37$ .

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

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

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