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Set of residues mod p closed with respect to the operation ab+1 (mod p)

Source: : 2nd Memorial Mathematical Competition "Aleksandar Blazhevski - Cane" - Problem 2

January 12, 2021
number theoryprime numbers

Problem Statement

Let pp be a prime number and F={0,1,2,...,p−1}F=\left \{0,1,2,...,p-1 \right \}. Let AA be a proper subset of FF that satisfies the following property: if a,b∈Aa,b \in A, then ab+1ab+1 (mod pp) ∈A \in A. How many elements can AA have? (Justify your answer.)
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