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 number of permutations of a set of $n$ elements is denoted by $P_n$.

Prove the equality $P_n=n!$.

For any positive integer $k$, the factorial $k!$ is defined as a product of all integers between 1 and $k$ inclusive: $k!=k\times (k-1)\times ...\times 1$. What’s the remainder when $2025!+2024!+2023!+...+3!+2!+1!$ is divided by $8$?