Question

prove that two consecutive number are always co-prime?

Answer 1

Let two consecutive numbers are nn and n+1n+1.

Assume they are not co-primes.

Then gcd(n,n+1)=xgcd(n,n+1)=x, because it can not equal to 11, xx is natural and x>1x>1
So xx divides nn as well as n+1n+1.

Then xx also divides n+1−nn+1−n, by general understanding.

Hence xx divides 11 or x=1x=1.

But we have assumed x>1x>1.

So by contradiction nn & n+1n+1 are co-prime.

Is it right or is there any better way to prove that two consecutive numbers are co-prime?