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if 19 divides a_1^9+a_2^9+...+a_9^9 then also divides a_1a_2...a_9

Source: 2013 Saudi Arabia Pre-TST 3.2

September 13, 2020
number theorydividesdivisible

Problem Statement

Let a1,a2,...,a9a_1, a_2,..., a_9 be integers. Prove that if 1919 divides a19+a29+...+a99a_1^9+a_2^9+...+a_9^9 then 1919 divides the product a1a2...a9a_1a_2...a_9.
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