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Let

n{}

be a natural number. The pairwise distinct nonzero integers

a_1,a_2,\ldots,a_n

have the property that the number

(k+a_1)(k+a_2)\cdots(k+a_n)

is divisible by

a_1a_2\cdots a_n

for any integer

k{}.

Find the largest possible value of

a_n.

Proposed by F. Petrov and K. Sukhov

Problem Statement