Question

prove that two consecutive positive integers are always coprime
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Answer 1

Step-by-step explanation:

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?

Answer 2

hey,,,

Let n and n+1 be two consecutive integers.

Let (n,n+1)=d

∴d∣nandd∣n+1

d∣(n+1)−nord∣1

∴d=1

∴(n,n+1)=1

i.e., n and (n+1) are relatively prime

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