Multiple of multinomial coefficient is an integer
Source: Romanian Master in Mathematics 2009, Problem 1
March 7, 2009
floor functioninequalitiesnumber theorygreatest common divisorleast common multipletriangle inequalityalgebra unsolved
Problem Statement
For a_i \in \mathbb{Z}^ \plus{}, i \equal{} 1, \ldots, k, and n \equal{} \sum^k_{i \equal{} 1} a_i, let d \equal{} \gcd(a_1, \ldots, a_k) denote the greatest common divisor of .
Prove that \frac {d} {n} \cdot \frac {n!}{\prod\limits^k_{i \equal{} 1} (a_i!)} is an integer.Dan Schwarz, Romania