MathDB
Divisibility

Source: APMO 2003

March 5, 2006
number theoryprime numbersAPMO

Problem Statement

Let k≥14k\ge 14 be an integer, and let pkp_k be the largest prime number which is strictly less than kk. You may assume that pk≥3k/4p_k\ge 3k/4. Let nn be a composite integer. Prove: (a) if n=2pkn=2p_k, then nn does not divide (n−k)!(n-k)!; (b) if n>2pkn>2p_k, then nn divides (n−k)!(n-k)!.
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