2023/026) Show that $n!+1$ and $(n+1)!+1$ are relatively prime for all natural n ?

We shall prove it by taking the GCD

$GCD((n+1)!+1, n!+1)$

$=  GCD((n+1)!+1- (n!+1), n!+1)$ using GCD(a,b) = GCD(a-mb,b) for any integer m

$=  GCD((n+1)!- n!, n!+1)$

$=  GCD((n!(n+1-1), n!+1)$

$=  GCD(n!.n, n!+1)$

$=  GCD(n!, n!+1)$ we can devide 1st term by n and GCD shall not change and 2nd term is not divisible by n

$=  GCD( n!, n!+1-n)$  using GCD(a,b) = GCD(a-mb,b) for any integer m

$=GCD(n!,1)= 1$ 

So these are relativvly primes