Nair's and Farhi's identities involving the least common multiple of binomial coefficients are equivalent

In 1982, Nair proved the identity: ${\rm lcm}({k \choose 1}, 2{k \choose 2}, >..., k{k \choose k}) ={\rm lcm}(1, 2, ..., k), \forall k\in \mathbb{N}.$ Recently, Farhi proved a new identity: ${\rm lcm}({k \choose 0}, {k \choose 1}, >..., {k \choose k}) =\frac{{\rm lcm}(1, 2, ..., k+1)}{k+1}, \forall k\in \mathbb{N}.$ In this note, we show that Nair's and Farhi's identities are equivalent.