Proving that $2^{2018}-2^{2016}+2^{2013}$ is divisible by $5$.

Question

Prove that $2^{2018}-2^{2016}+2^{2013}$ is divisible by $5$.

I'm not sure how do I even start, so may I get help please?

Answer 1

Simply factorise:

$2^{2018}-2^{2016}+2^{2013}=2^{2013}(2^5-2^3+1)=2^{2013}(25)$

Clearly $5 \mid 25$, so you are done.

Answer 2

Hint

$$2^2\equiv -1\mod 5$$ so, raise it to suitable powers to conclude the result.