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Show that n | u_{n, k} for all n and 1 ≤ k ≤ n

Source: IMO LongList 1982 - P38

March 20, 2011
symmetrynumber theory proposednumber theory

Problem Statement

Numbers un,k (1≤k≤n)u_{n,k} \ (1\leq k \leq n) are defined as follows u_{1,1}=1,   u_{n,k}=\binom{n}{k} - \sum_{d \mid n, d \mid k, d>1} u_{n/d, k/d}. (the empty sum is defined to be equal to zero). Prove that n∣un,kn \mid u_{n,k} for every natural number nn and for every k (1≤k≤n).k \ (1 \leq k \leq n).
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