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2^{3k} divides difference of binomials but not 2^{3k+1}

Source: IMO Shortlist 2007, N4, AIMO 2008, TST 6, P2

July 13, 2008
binomial coefficientsnumber theoryDivisibilityIMO ShortlistPolandHi

Problem Statement

For every integer k≥2, k \geq 2, prove that 23k 2^{3k} divides the number \binom{2^{k \plus{} 1}}{2^{k}} \minus{} \binom{2^{k}}{2^{k \minus{} 1}} but 2^{3k \plus{} 1} does not.
Author: Waldemar Pompe, Poland
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