Question

For which other positive integers a, less than 11, will the number (a^n) + (a^n+1) + (a^n+2) + (a^n+3) + (a^n+4) always be divisible?
Pls, Answer with the entire explanation.

Answer 1

If the number is supposed to be

$a^n + a^{n+1} + a^{n+2} + a^{n+3} + a^{n+4}$

then it can be factorized as

$a^n \left(1 + a + a^2 + a^3 + a^4\right)$

but there's not much to say about divisibility here without any more information about a.

If you meant

$a^n + (a^n+1) + (a^n+2) + (a^n+3) + (a^n+4)$

simplifying gives

$5a^n + 10 = 5 (a^n+2)$

which is clearly divisible by 5.