Final answer:
The gcd of expressions of the form 3n+1 and n, as well as an+1 and n, is 1. This can be proven using the Euclidean algorithm.
Step-by-step explanation:
The given question is about the greatest common divisor (gcd) of two numbers. The specific problem is to find the gcd of the expressions 3n+1 and n, as well as an expression of the form an+1 and n, where a is a positive integer.
Let's start by finding the gcd of 3n+1 and n. To do this, we can use the Euclidean algorithm. We divide 3n+1 by n, which gives us a quotient of 3 and a remainder of 1. We then divide n by 1, which gives us a quotient of n and no remainder. The gcd of 3n+1 and n is 1.
Similarly, for any positive integer a, we can find the gcd of an+1 and n. When we divide an+1 by n, we get a quotient of a and a remainder of 1. When we divide n by 1, we get a quotient of n and no remainder. So the gcd of an+1 and n is also 1.