Problems

Age
Difficulty
Found: 5

Let us introduce the notation – we denote the product of all natural numbers from 1 to n by n!. For example, 5!=1×2×3×4×5=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 2015000000, where the number of 0’s at the end can be arbitrary?

The numbers 1,2,,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.

The number of permutations of a set of n elements is denoted by Pn.

Prove the equality Pn=n!.

For any positive integer k, the factorial k! is defined as a product of all integers between 1 and k inclusive: k!=k×(k1)×...×1. What’s the remainder when 2025!+2024!+2023!+...+3!+2!+1! is divided by 8?