Problem

Source: Turkey 2021 IMO TST Problem 9

Tags: Turkey, number theory, GCD



For which positive integer couples (k,n), the equality \Bigg|\Bigg\{{a \in \mathbb{Z}^+: 1\leq a\leq(nk)!, gcd \left(\binom{a}{k},n\right)=1}\Bigg\}\Bigg|=\frac{(nk)!}{6} holds?