Question

prove that two consecutive integers are always coprime​

Answer 1

Answer:

Let two consecutive numbers are n and n+1.

Assume they are not co-primes.

Then gcd(n,n+1)=x, because it can not equal to 1, x is natural and x>1

So x divides n as well as n+1.

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

Hence x divides 1 or x=1.

But we have assumed x>1.

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

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

second method.---

Your proof looks good. Using the method of contradiction is not a bad idea here but you could have skipped that in your prove.

Given that n and n+1 are two consecutive integers. Now suppose gcd(n,n+1)=p. Then p|n and p|n+1. Which implies that p|n+1−n or p|1. There is no number which divides 1 except 1. So p=1 or you can say gcd(n,n+1)=p=1. Which implies n and n+1 are coprime.

I think it's help you...