Is 7^2015 + 4^2015 divisible by 17? Explain your reasoning and show your work.

Question

Is $7^{2015} + 4^{2015}$ divisible by 17? Explain your reasoning and show your work.

I'm confused on how exactly I would do this. Would I need to use Fermats Theorem?

Answer 1

Using Fermat's little theorem and the fact that 16 divides 2016, the given expression (mod 17) is equal to $4^{-1}+7^{-1}$ (mod 17) which is $13+5=18=1$ (mod 17); alternatively, $4 \neq -7$ (mod 17) is sufficient to prove that the given expression is not divisible by 17.

Answer 2

Would I need to use Fermat's Theorem ?

Not necessarily; for instance, you could simply write $7^2=49=51-2=3\cdot17-2$, along with $(-2)^4=4^2=16=17-1$.