Consecutive integers

Number Theory Level 3

What is the smallest positive integer that can be expressed as the sum of nine consecutive integers, the sum of ten consecutive integers and the sum of eleven consecutive integers?

The answer is 495.

1 solution

Chew-Seong Cheong
Jul 8, 2015

Let the required smallest positive integer by $n$ and the smallest integer of the nine, ten and eleven consecutive integers be $a$ , $b$ and $c$ respectively. Then, we have:

$n = \begin{cases} \dfrac{9(2a+8)}{2} & = 9a + 36 & \equiv 0 \pmod{9} \\ \dfrac{10(2b+9)}{2} & = 10b + 45 & \equiv 0 \pmod{5} \\ \dfrac{11(2c+10)}{2} & = 11a + 55 & \equiv 0 \pmod{11} \end{cases}$

Therefore, $n$ is divisible by $5$ , $9$ and $11$ and $n = LCM(5,9,11) = \boxed{495}$ .

Moderator note:

Great solution.

This was one of our problems from the distant past , and I'm excited to see it pop up again.

Consecutive integers

The answer is 495.

1 solution

Moderator note:

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