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f(f(x))-4x is divisible by 17 for all integers 1 <= x <= 16?

Source: 2021 Dürer Math Competition Finals Day2 E14 https://artofproblemsolving.com/community/c2749870_

January 8, 2022
number theorydivisibledivides

Problem Statement

How many functions f:{1,2,...,16}→{1,2,...,16}f : \{1, 2, . . . , 16\} \to \{1, 2, . . . , 16\} have the property that f(f(x))−4xf(f(x))-4x is divisible by 1717 for all integers 1≤x≤161 \le x \le 16?
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