# Math Help - Fermat's Little Theorem

1. ## Fermat's Little Theorem

I want to use Fermat's Little Theorem to deduce that

$13^{16n+2} + 1$

is divisible by 7, where n is a positive integer.

Does this have anything to do with modular arithmetic? D:
If so, would I set it up like $13^{16n+2} \equiv 1 (mod 7)$

2. Hello,
Originally Posted by RAz
I want to use Fermat's Little Theorem to deduce that

$13^{16n+2} {\color{red}-} 1$

is divisible by 7, where n is a positive integer.

Does this have anything to do with modular arithmetic? D:
If so, would I set it up like $13^{16n+2} \equiv 1 (mod 7)$
You made a typo... it's a minus sign, not a + sign.

Anyway, you don't need to use Fermat's little theorem.

Indeed $13\equiv -1 (\bmod 7) \Rightarrow 13^2 \equiv 1(\bmod 7)$

Hence $13^{16n+2}=(13^{2(8n+1)})=(13^2)^{8n+1}\equiv 1(\bmod 7) \quad \square$