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Gcd(3n+1,n)=1, and in fact that gcd(an+1,n)=1 when a is a positive integer.

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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.

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