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

Sol. 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 i.e. n = 97
now 96! + 1 and 97! are both divisible by 97
so HCF = 97
(by Wilson's theorem (p – 1)! + 1) is divisible by p)