Prove that for any integer n the integers 3n 1 and 2n 1 ar

Prove that, for any integer n, the integers 3n + 1 and 2n + 1 are relative primes.

Two numbers are said to be relative prime numbers when they have a common factor as 1.

Relatively prime numbers have greatest common factor(GCF) of 1.

In the given problem 3n+1 and 2n+1 are the given numbers where n is an integer.

We need to prove that these two numbers are relative primes.

Integers can be positive as well as negative.

Let us substitute few positive numbers.

Let n=1

Therefore, the two numbers are

3n+1=3(1) +1=3+1=4

2n+1=2(1) +1=2+1=3

The GCD of these 4 and 3 is 1. Therefore, they are relatively prime.

Let us consider another number where

Let n=3

Therefore, the two numbers are :

3n+1=3(3) +1=9+1=10

2n+1=2(3) +1=6+1=7

The GCD of these 10 and 7 is 1. Therefore, they are relatively prime.

Let us substitute few negative numbers.

Let n=-1

Therefore, the two numbers are:

3n+1=3(-1) +1=-3+1=-2

2n+1=2(-1) +1=-2+1=-1

The GCD of these -2 and -1 is 1. Therefore, they are relatively prime.

Let us consider another number where

Let n=-3

Therefore, the two numbers are:

3n+1=3(-3) +1=-9+1=-8

2n+1=2(-3) +1=-6+1=-5

The GCD of these -8 and -5 is 1. Therefore, they are relatively prime.

Hence, Its proved that for any integer n.

3n+1 and 2n+1 are relative primes.