Question 174
Let $k$ and $m$ be integers with $1 \lt k \lt m$. For a positive integer $i$, let $L_i$ be the least common multiple of $1, 2, \dots, i$. Prove that $k$ is a divisor of $L_i \cdot \left[\binom{m}{i} - \binom{m-k}{i}\right]$ for all $i \ge 1$. [Here, $\binom{n}{i} = \frac{n!}{i!(n-i)!}$ denotes a binomial coefficient. Note that $\binom{n}{i} = 0$ if $n \lt i$.]