Problems

Let us introduce the notation – we denote the product of all natural numbers from 1 to $n$ by $n!$. For example, $5!=1\times 2\times 3\times 4\times 5=120$.

a) Prove that the product of any three consecutive natural numbers is divisible by $3!=6$.

b) What about the product of any four consecutive natural numbers? Is it always divisible by 4!=24?

Is it possible for $n!$ to be written as $2015000\dots 000$, where the number of 0’s at the end can be arbitrary?

The numbers $1,2,\dots ,9$ are divided into three groups. Prove that the product of the numbers in one of the groups will always be no less than 72.