Number theory · Modular arithmetic · Powers and residues · Order of a residue
Problem 1, 2018
Let \(a\), \(b\) and \(c\) be positive integers for which both of the numbers
\[ 24^{a} + 2^{b} + 2018^{c} \qquad \text{and} \qquad 10^{c} + 3^{a} + 2018^{b} \]
are divisible by \(7\). Prove that the number \(30^{b} + 3^{c} + 2018^{a}\) is not divisible by \(7\).
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Serbian Regional Competition 2018, high school grade I, category A, problem 1. Organized by the Mathematical Society of Serbia (DMS). Source