Same Remainder Every Time

Number Theory Level 1

If $n$ is a positive integer, what is the remainder when $\large n \times (n+1) \times (2n+1)$ is divided by 6?

3 2 5 1 4 0

1 solution

Ossama Ismail
Feb 9, 2016

The sum of the first n squares, $1^2 + 2^2 + ... + n^2 = n(n + 1)(2n + 1) / 6$ .

If $n$ and $(n + 1)$ are consecutive integers then exactly one of them will be even and divisible by 2. If neither $n$ nor $(n + 1)$ is divisible by $3$ then $(n + 2)$ will be. Therefore $n(n + 1)(2n + 1)$ is divisible by both 2 and 3.

7l momtaz :)

Refaat M. Sayed - 5 years, 4 months ago

Same Remainder Every Time

1 solution

0 pending reports