# Math Help - divisible

1. ## divisible

Prove that

$1^{1987} + 2^{1987} + ... + 13^{1987}$

is divisible by $15$.

2. Hello,
Originally Posted by perash
Prove that

$N=1^{1987} + 2^{1987} + ... + 13^{1987}{\color{red}+14^{1987}}$

is divisible by $15$.
I guess there is a typo.. (in red)
$14 \equiv -1 (\bmod 15)$
$13 \equiv -2 (\bmod 15)$
$12 \equiv -3 (\bmod 15)$
$11 \equiv -4 (\bmod 15)$
etc...

So $N=1^{1987} + 2^{1987} + ... + 13^{1987}+14^{1987} \equiv 1^{1987} + 2^{1987}+3^{1987}$ $+4^{1987}+5^{1987}+6^{1987}+7^{1987}-7^{1987}-6^{1987}-5^{1987}-4^{1987}-3^{1987}-2^{1987} -1^{1987} (\bmod 15)$

$N \equiv 0 (\bmod 15)$