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\sum_{m \in M} (-1) ^m (n m) is divisible by p^q

Source: 12th or 13th QEDMO problem 2 (11. - 15. 12. 2013) https://artofproblemsolving.com/community/c2400093_2013_qedmo_13th_or_12th

July 5, 2021
Binomialnumber theorydivisibledivides

Problem Statement

Let pp be a prime number and n,kn, k and qq natural numbers, where q≤n−1p−1q\le \frac{n -1}{p-1} should be. Let MM be the set of all integers mm from 00 to nn, for which m−km-k is divisible by pp. Show that ∑m∈M(−1)m(nm)\sum_{m \in M} (-1) ^m {n \choose m} is divisible by pqp^q.
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