Powers of odd numbers

Consider the polynomial $P(t) = t^5 + qt^4 + rt^3 + st^2 + ut + v$ with roots $a, b, c, d, e$ . The condition from the statement implies that $q$ is divisible by $n$ . Moreover, since

$\large\ \displaystyle \sum {ab} = \frac {1}{2} (\sum {a})^2 - \frac {1}{2}(\sum {a^2}),$

it follows that $r$ is also divisible by $n$ . Adding the equalities $P(a) = 0, P(b) = 0, P(c) = 0, P(d) = 0, P(e) = 0,$ we deduce that

$\large\ a^5 + b^5 + c^5 + d^5 + e^5 + s(a^2 + b^2 + c^2 + d^2 + e^2) + u(a + b + c + d + e) + 5v$

is divisible by $n$ .

But since $v = -abcde$ , it follows that $a^5 + b^5 + c^5 + d^5 + e^5 - 5abcde$ is divisible by $n$ , and we are done.