# If the product of two consecutive integers is 182, how do you find the integers?

Jun 28, 2015

The positive solutions are:

$\left\lfloor \sqrt{182} \right\rfloor = 13$ and $\left\lceil \sqrt{182} \right\rceil = 14$

There are also negative solutions: $- 14$ and $- 13$

#### Explanation:

If the positive integers are $n$ and $n + 1$, then

${n}^{2} < n \left(n + 1\right) < {\left(n + 1\right)}^{2}$

So

$n = \sqrt{{n}^{2}} < \sqrt{n \left(n + 1\right)} < \sqrt{{\left(n + 1\right)}^{2}} = n + 1$

Hence

$n = \left\lfloor \sqrt{n \left(n + 1\right)} \right\rfloor$ and $n + 1 = \left\lceil \sqrt{n \left(n + 1\right)} \right\rceil$