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T(n,a_1,a_2,...,a_{15}) = (a_1^n+a_2^n+ ...+a_{15}^n)a_1a_2...a_{15}

Source: Israel Grosman Memorial Mathematical Olympiad 1999 p1

February 15, 2020

divisiblenumber theory

Problem Statement

For any

16

positive integers

n,a_1,a_2,...,a_{15}

we define

T(n,a_1,a_2,...,a_{15}) = (a_1^n+a_2^n+ ...+a_{15}^n)a_1a_2...a_{15}

. Find the smallest

n

such that

T(n,a_1,a_2,...,a_{15})

is divisible by

15

for any choice of

a_1,a_2,...,a_{15}