Relatively Prime

Two numbers are said to be relatively prime when they have only 1 as the common factor or we can say that there is no same value other than one that you could divide them both and get zero as a remainder.

If the only common factor of two numbers a and b is 1, then a and b are relatively prime numbers. In this case, (a, b) is said to be a relatively prime pair. These numbers need not be prime numbers always. Two composite numbers can also be relatively primes, for example, 9 and 10. Relatively prime numbers are also referred to as mutually prime (or) coprime numbers.

To find whether two numbers are relatively prime or not, we find the HCF of the numbers. If the HCF is 1, then the two numbers are said to be relatively primes. The HCF-Highest Common Factor of two numbers can be found by listing down all the factors and then selecting the highest common factor out of those.

For example: Let's determine whether the given pairs of numbers are relatively prime or not a) 7 and 9; b) 8 and 10.
a) 7 and 9
The factors of 7 are 1 and 7
The factors of 9 are 1, 3, and 9
1 is the only common factor of 7 and 9
HCF of (7, 9) = 1
Thus, (7, 9) is relatively prime because only 1 is a common factor.

b) 8 and 10
The factors of 8 are 1, 2, 4, and 8
The factors of 10 are 1, 2, 5, and 10
1 and 2 are the common factors of 8 and 10.
HCF (8, 10) = 2
Thus, (8,10) is not relatively prime.

Some of the properties of relatively prime numbers are:

• The HCF-Highest Common Factor of two relatively prime numbers is always 1. For example, 5 and 9 are relatively prime numbers, and hence, HCF (5, 9)= 1
• The LCM-Least Common Multiple of two relatively prime numbers is always their product. The relation between HCF and LCM of two numbers, suppose a and b, is HCF (a, b) × LCM (a, b) = a × b. As the numbers are relatively prime their HCF is 1 therefore the product of numbers is equal to the LCM of numbers. For example, 2 and 3 are relatively prime numbers. Hence, LCM = 2 × 3 = 6
• The sum of two relatively prime numbers is always relatively prime with their product. For example, 2 and 3 are relatively prime numbers. Here, 2 + 3 = 5 is relatively prime with 2 × 3 = 6. The only common factor of 5 and 6 is 1.
• Any two prime numbers are always relatively prime. For example, in 19 and 17 the only common factor is 1 and they are prime numbers too.
• A prime number is relatively prime with any other number because prime numbers are the numbers that can be divided by one or themselves. Thus, if we pair up any prime number with other numbers the result will be relatively prime because the common factor will be one. For example, 17 and 25 are relatively prime because the common factor of both numbers is 1. Factors of 17 are 1 and 17 and factors of 25 are 1, 5, 25.
• Any two consecutive numbers are always relatively prime. For example, 28 and 29 are two consecutive numbers. Factors of 28 are 1, 2, 4, 7, 14, 28, and factors of 29 are 1, 29, so the only common factor is 1.

Important Notes

Given below are some of the important notes related to relatively prime that we read in this article. Have a look!

• Any two prime numbers are relatively prime.
• Any two consecutive numbers are relatively prime.
• A prime number is relatively prime with any other number.
• Two even numbers are NEVER relatively prime because number 2 is a factor of all even numbers. Hence they are not relatively prime.
• Two numbers are relatively prime if their HCF is 1 and vice versa.
• Relatively prime numbers don't need to be prime numbers. For example, two composite numbers 12 and 35 are relatively prime numbers because their HCF is 1.

Related Articles on Relatively Primes

Check out the important topics mentioned below to learn more about the relatively prime numbers and its related topics.

• Prime Numbers
• Prime Factorization
• Coprime Numbers

What are Relatively Prime Numbers?

Two numbers are said to be relatively prime if the common factor between the numbers is one. For example, 34 and 35 are relatively prime. Factors of 34 are 1, 2, 17, 34, and factors of 35 are 1, 5, 7, 35. the HCF-Highest Common Factor is one.

Are 4 and 9 Relatively Prime?

Yes, 4 and 9 are relatively prime because the HCF of 4 and 9 is 1. Factors of 4 are 1,2, 4, and factors of 9 are 1,3,9.

Is 42 and 77 Relatively Prime?

No, 42 and 77 are not relatively prime because the HCF of 42 and 77 is 7. Factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42, and factors of 77 are 1, 7, 11, and 77. Thus we can see that 7 is the HCF of 42 and 77.

How do you Prove Two Consecutive Numbers are Relatively Prime?

Let's assume n and n+1 are two consecutive numbers. If not a single factor of n is similar to factors of n+1 except one. That means the only common factor of two consecutive numbers is 1. Therefore, n and n+1 are relatively prime. For example, 39 and 40 are two consecutive numbers. Factors of 39 are 1, 3, 13, 39, and factors of 40 are 1, 2, 4, 5, 8, 10, 20, 40. We can see that only 1 is common in the factors of two consecutive numbers, therefore 39 and 40 are relatively prime.

Which Positive Integers less than 12 are Relatively Prime to 12?

The positive integers less than 12 that are relatively prime to 12 are 1, 5, 7, and 11.