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divisibility with 13

Source: Romania Nationals RMO 2005 - grade 7, problem 2

March 31, 2005
modular arithmetic

Problem Statement

Let a,ba,b be two integers. Prove that a) 13∣2a+3b13 \mid 2a+3b if and only if 13∣2b−3a13 \mid 2b-3a; b) If 13∣a2+b213 \mid a^2+b^2 then 13∣(2a+3b)(2b+3a)13 \mid (2a+3b)(2b+3a). Mircea Fianu
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