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## Not square-free numbers

Suppose that \(m\) is a positive integer that is not square-free. Show that there exist integers \(a_1\) and \(a_2\) such that \(a_1 \not\equiv a_2 (\mbox{mod}\ m)\), but \(a_1^k \equiv a_2^k (\mbox{mod}\ m)\) for all integers \(k > 1\).

**Source:**from book "An Introduction to the Theory of Numbers"