Can anyone help me in proving that two consecutive numbers are co-primes?
My approach:
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 $x\gt1$
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 $x\gt 1$.
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?
Best Answer
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.
Notice that I have not used contradiction anywhere.