To show that (n, n+1)= 1, suppose it is not. Suppose (n, n+1)= m> 1. Then m is a common factor of n and n+1 and we must have n= mk and n+1= mp for integers k and p. Then n+ 1= mk+ 1= mp so 1= mp-mk= m(p- k). But, since the only factor of 1 is 1 itself, we must have both m and p-k equal to 1 and m= 1 contradicts m> 1.