Question

For each positive integer n, consider the highest common factor hn of the two numbers n! + 1 and (n + 1)!.
For n < 100, find the largest value of hn.

Answer 1

Answer:

n ! + 1 is not divisible by 1, 2, ….., n

(n + 1) ! is divisible by 1, 2, …., n

so HCF  n + 1

also (n + 1) ! is not divisible by n + 2, n + 3, ….

so HCF can be (n + 1) only

let us start by taking n = 99

 99 ! + 1 and 100 !

HCF = 100 is not possible as 100 divides 99 !

composite number will not be able to make it

so let us take prime number i.e. n = 97

now 96 ! + 1 and 97 ! are both divisible by 97

so HCF = 97

by Coilson theorem ((p – 1) ! + 1) is divisible by p.

HOPE IT'S ENOUGH.........