# Which of the following numbers is not the sum of three consecutive integers: 51, 61, 72, 81?

Oct 19, 2016

$61 \text{ }$ it is the only one not divisible by 3.

#### Explanation:

One of the properties of any three consecutive numbers is that their sum is always a multiple of 3.

Why is this?

Consecutive numbers can be written as $x , x + 1 , x + 2 , x + 3 , \ldots$

The sum of 3 consecutive numbers is given by

$x + x + 1 + x + 2$ which simplifies to

$3 x + 3$

=$\textcolor{red}{3} \left(x + 1\right)$

The $\textcolor{red}{3}$ shows that the sum will always be a multiple of 3.

Which of the given numbers are divisible by 3?

You can simply add their digits to find out.
If the sum of the digits of a number is a multiple of 3, then the number itself is divisible by 3.

$51 : 5 + 1 = 6 \text{ }$ 51 is divisible by 3
$61 : 6 + 1 = 7 \text{ }$ 61 is not divisible by 3,
$72 : 7 + 2 = 9 \text{ }$ 72 is divisible by 3
$81 : 8 + 1 = 9 \text{ }$81 is divisible by 3

Only 61 is not divisible by 3. Therefore it is not the sum of three consecutive numbers.