Divisibility involving root of unity

Let $p$ be a prime number and $\omega$ be a $p$-th root of unity. Suppose $a_0,a_1, \dots, a_{p-1}, b_0, b_1, \dots, b_{p-1}$ be integers such that $a_0 \omega^0+a_1 \omega^1+ \dots a_{p-1} \omega^{p-1}$ and $b_0 \omega^0 + b_1 \omega^1 + \dots b_{p-1} \omega^{p-1}$ are also integers

Prove that $(a_0 \omega^0+a_1 \omega^1+ \dots a_{p-1} \omega^{p-1})-(b_0 \omega^0 + b_1 \omega^1 + \dots b_{p-1} \omega^{p-1})$ is divisible by $p$ if and only if $p$ divides all of $a_0-b_0$, $a_1-b_1$, $\dots$, $a_{p-1}-b_{p-1}$