ZA-SAMO 2016 Problem 6

Let $k$ and $m$ be integers with $1 < k < m$ . For a positive integer $i$ , let $L_i$ be the least common multiple of $1,2,\ldots,i$ . Prove that $k$ is a divisor of $L_i \cdot [\binom{m}{i} - \binom{m-k}{i}]$ for all $i \geq 1$ . [Here, $\binom{n}{i} = \frac{n!}{i!(n-i)!}$ denotes a binomial coefficient. Note that $\binom{n}{i} = 0$ if $n < i$ .]