Question

Prove that gcd(n, n 1) = 1 for all positive integers n?

Answer 1

To prove gcd(n, n+1) = 1 for all positive integers n, we can use the Euclidean algorithm. Assume d is the gcd of n and n+1. By subtracting the equations, we find that d divides 1. Since 1 is a prime number, the only divisors are 1 and -1. Therefore, d must equal 1.

Step-by-step explanation:

To prove that gcd(n, n+1) = 1 for all positive integers n, we need to show that the greatest common divisor of any positive integer n and its immediate successor n+1 is always 1.

We can use the Euclidean algorithm to prove this. Let's assume that d is the greatest common divisor of n and n+1. According to the Euclidean algorithm, we have:

1. d divides n, which means n = dx for some positive integer x.
2. d divides n+1, which means n+1 = dy for some positive integer y.
3. Subtracting equation (1) from equation (2), we get 1 = dy - dx = d(y - x).
4. This implies that d divides 1, and since 1 is a prime number, the only divisors of 1 are 1 and -1. Therefore, d must equal 1.

Therefore, gcd(n, n+1) = 1 for all positive integers n.