Školjka

a) Prove that for very positive integer $m$ there exists an integer $n \geq m$ such that $$\bigg\lfloor\frac{n}{1}\bigg\rfloor\cdot\bigg\lfloor\frac{n}{2}\bigg\rfloor\cdot\ldots\cdot\bigg\lfloor\frac{n}{m}\bigg\rfloor = \binom{n}{m}$$ b) Denote by $p(m)$ the smallest integer $n \geq m$ such that the above equation holds. Prove $p(2018) = p(2019)$.