An identity involving the least common multiple of binomial coefficients and its application

In this paper, we prove the identity $$\lcm\{\binom{k}{0}, \binom{k}{1}, >..., \binom{k}{k}\} = \frac{\lcm(1, 2, ..., k, k + 1)}{k + 1} (\forall k \in \mathbb{N}) .$$ As an application, we give an easily proof of the well-known nontrivial lower bound $\lcm(1, 2, ..., k) \geq 2^{k - 1}$ $(\forall k \geq 1)$.