mathwonk said:the more trivial the inquiry the more replies.
The GCD of (n, n+1) is 1.
There are several methods to prove the GCD of (n, n+1) is 1. One method is to use the Euclidean algorithm, which involves repeatedly dividing the larger number by the smaller number and taking the remainder until the remainder is 0. The last non-zero remainder will be the GCD. In the case of (n, n+1), the remainder will always be 1, proving that the GCD is 1.
Proving the GCD of (n, n+1) is 1 is significant because it shows that n and n+1 are relatively prime, meaning they have no common factors other than 1. This has applications in number theory and cryptography.
No, the GCD of (n, n+1) can only be 1. This is because n and n+1 are consecutive integers, and the only common factor they can have is 1.
Proving the GCD of (n, n+1) is 1 is related to the concept of coprime numbers, which are numbers that have a GCD of 1. It is also related to the fundamental theorem of arithmetic, which states that every positive integer can be uniquely expressed as a product of prime numbers. In the case of (n, n+1), n and n+1 cannot share any prime factors other than 1, further supporting the fact that their GCD is 1.