I had this questions from a previous exam that I couldn't answer, I am apologizing for any English mistakes or for any stupid questions, I tried to solve them and I searched the internet and I couldn't find answers.
3-if $\ \ \ a \in [1,10] \ \ a \ is \ an \ integer$ $$a^{2016}+a^{2017} = ?$$ For how many values of a is the result divisble by 5 ?
*I tried to take $a^{2016}$ common but I was left with $a^{2016}(1+a)$ so if a is equal to 4 the whole thing would be multiplied by 5 which would make it divsible by 5 and if a is equal to 9 too but I couldn't figure it out for the other numbers.
Thanks for taking the time to read the question !
You had the right idea.
$$a^{2016}(a+1)$$ is divisble by $5$ if and only if either $a$ is divisble by $5$ or $a+1$ is divisble by $5$. The rest should be easy.