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

Step-by-step solution

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 )