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.