Number theory · Divisibility · Modular arithmetic · Powers of two · Order of an element
Let \(k\) be a positive integer. Prove that the number
\[ 2^{2k-1} + 2^{k} + 1 \]
is never divisible by \(7\).
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Serbian Regional Competition (Okruzno takmicenje) 2003, high school grade I, category A, problem 1. Organized by the Mathematical Society of Serbia (DMS). Source