# What are Relatively Prime Numbers

The greater common factor of the pair of numbers is the largest number that evenly divides into both numbers. If two numbers are relative prime then their greater common factor will be one. If two numbers are not relative prime then their greater common factor will be greater than one. Relative prime also termed as coprime. If two numbers are said to be relatively prime, then the only positive common divisor will be 1.

Two numbers are relativity prime, doesn't imply that the same numbers are prime numbers.
For example, 35 and 18 are relatively prime, but both are not prime numbers.

## What are Relative Prime Numbers

Two integers are said to be relatively prime if the only positive integer that divides both of them evenly is 1. Relatively prime integers are sometimes also called strangers or coprime. In other words, relatively prime numbers have a greatest common factor of 1. Two integers are said to be relatively prime if they do not have any common factor other than 1.

Two integers a and b whose greatest common divisor is equal to one are called relative prime,

that is  gcd(a, b) = 1.

For example: gcd(2, 3) = 1

=> 2 and 3 are relative prime.

## Properties of Relatively Prime Numbers

The following are some properties of relatively prime numbers:
One is relatively prime with all other numbers.
For example, 3 and 1 are relatively prime.

Any two prime numbers are always relatively prime.
For example, lets consider two prime numbers 11 and 19.

=> gcd(11, 19) = 1

=> 11 and 19 are relatively prime to each other.

If $a$ and $b$ are relatively prime, then $2a - 1$ and $2b - 1$ are also relatively prime.
For example, Let a = 23 and b = 7 are relatively prime.

Now $2a - 1 = 2 * 23 - 1 = 46 - 1 = 45$

$2b - 1 = 2 * 7 - 1 = 14 - 1 = 13$

=> 45 and 13 are also relatively prime.

Two consecutive integers will always be relatively prime.
For example, lets consider 12 and 13 be consecutive numbers

=> gcd(12, 13) = 1

=> 12 and 13 are relatively prime

Sum of any two relatively prime numbers will be relatively prime to their products.
Example, Lets consider two relativity prime numbers 3 and 5.

Sum of numbers = $3 + 5 = 8$

Product of numbers = $3 * 5 = 15$

For 8 and 15, the only common term is 1. So 8 and 15 are relatively prime.

## Relative Prime Numbers

For any two integers, the greatest common factor of both of them will be 1, that is g.c.d = 1. So both the numbers together will not have any common factors other than 1. All natural numbers greater than 1 can be classified as either prime or composite. Beside looking at individual numbers also look at at pair of numbers. A pair of numbers is relative prime if the only common factor of the two numbers is one.

## Solved Examples

Question 1: Is 15 and 18 are relative prime?
Solution:
Given numbers are 15 and 18

=> The common factor of the two numbers = 1

or gcd(15, 18) = 1

=> Yes, 15 and 18 are relative prime.

Question 2: Check whether 33 and 40 are relative prime or not.

Solution:
Step 1: Find the factors of both the numbers

The factors of 33 = 1, 3, 11 and 33

The factors of 40 = 1, 2, 4, 5, 8, 10, 20 and 40

Step 2: Find GCD

The only common factor of 33 and 40 is 1

So 33 and 40 are relatively prime.

Question 3: Choose the pair of relatively prime numbers in the following.
A.  (7, 15)
B.  (2, 16)
C.  (6, 12)
D.  (5, 15)
Solution:
Step 1: Common factor of 7 and 15 is only 1.

Step 2: Common factor of 2 and 16 are 1 and 2.

Step 3: Common factor of 6 and 12 are 1, 2, 3 and 6.

Step 4: Common factor of 5 and 15 are 1 and 5.

=> we know, two numbers are said to be relatively prime if their common factor is only 1

=> Common factor of 7 and 15 = 1

=> Option (A) is the correct answer.