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n! divides product of (2^n-2^k) for k=0,..,n-1

Source: Turkish NMO 1996, 5. Problem

July 31, 2011
group theoryabstract algebrafloor functionmodular arithmeticnumber theory proposednumber theory

Problem Statement

Prove that ∏k=0n−1(2n−2k)\prod\limits_{k=0}^{n-1}{({{2}^{n}}-{{2}^{k}})} is divisible by n!n! for all positive integers nn.
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