# If the sum of two consecutive integers is less than 83, how do you find the pair of integers with the greatest sum?

Jun 4, 2015

Two integers can only add to become another integer. The greatest integer less than $83$ is $82$.

However, you can't have two consecutive integers adding up to give an even number, so we're going to have to pick the greatest odd number that's smaller than $83$, which is of course $81$.

So we are looking for two consecutive integers that add up to $81$. We are looking for a number $a$ and the number after it $a + 1$.
To do this, we can add them and set them equal to $83$.

$a + \left(a + 1\right) = 83$
$2 a = 82$

$a = 41$
$\left(a + 1\right) = 42$
$41 + 42 = 83$

And that is good information, by we are looking for the biggest odd number that's smaller than $83$ so we can subtract $1$ from both numbers.

$\left(a - 1\right) = 40$
$a = 41$
$40 + 41 = 81$