Let $\mathcal{Q}=19^5+660^5+1316^5$ . We know that $25$ is a factor of $\mathcal{Q}$ . Find with a proof the largest prime factor of $\mathcal{Q}$ not exceeding of $10,000$.
I found by computer search that the answer is $19$ but would like to see a more mathematical proof.
$660+1316$ divided by 19.
I hope it will help.
We know that $a^5+b^5=(a+b)(a^4-a^3b+a^2b^2-ab^3+b^4)$.
Thus, our number divided by $19$.
By the same we we can get that our number divided by $7$ and by $3$,
but it's not so relevant here.