In the number system theory, numbers are categorised based on their properties. Today, we are going to learn about one such categorization called coprime numbers. But before we delve deep into it, let us first understand the concept of prime numbers. Prime numbers refer to:

Natural numbers greater than 1

Which have only two divisors, those being 1 and the number itself

In other words, those numbers that cannot be written as the product of two smaller natural numbers

Now, coming towards our main question, what is the definition of coprime numbers?

‘Co’ refers to a ‘pair’. So, we can define a pair of integers, let's say a and b, as coprime or primes to each other or mutually prime or relatively prime if:

the only positive integer that divides both of them to give a remainder zero is 1

In other words, the HCF of the two numbers is 1

Coprime numbers are denoted as gcd(a, b)=1 or (a, b)=1 or as “a is prime to b”

Some of the pairs of coprime numbers from 1 to 100 are (2,3), (3,5), (5,7), (11,13), (17,19), (21,22), (29,31), (41,43), (59,61), (71,73), (87,88), (99,100)

So this was the basic difference between the prime and the coprime numbers. Moving towards our next topic, we have:

How to find Coprime numbers?

Let us understand coprime numbers with the help of a detailed example which will also clarify how we can find them?

Example 1

Are 6 and 15 coprime numbers?

Solution

Factors of 6 = 1, 2, 3, 6

Factors of 15 = 1, 3, 5, 15

On comparing the factors of these two numbers 6 and 15, we see that both these numbers have factors 1 and 3 in common. The HCF for this pair is 3. Therefore the pair is not coprime.

Pair is not coprime

Example 2

Are 10 and 21 coprime numbers?

Solution

For the given two numbers, find out what their factors are

Factors of 10 = 1, 2, 5, 10

Factors of 21 = 1, 3, 7, 21

On comparing the factors of these two numbers 10 and 21, we see that both these numbers have only factor 1 in common. The HCF for this pair is 1. Therefore the pair is coprime.

Pair is coprime

Properties of Coprime numbers

Numbers that are coprime may not necessarily be prime numbers themselves. For eg: (10,21), the numbers are not prime numbers if taken individually but they are coprime if considered as a pair. This concludes that composite numbers can be coprime numbers too.

Prime numbers are coprime by default. For eg: (2,3), (5,11), (19,23) and so on. Every prime number has only two factors i.e. 1 and the number itself. Therefore, the only common factor of a pair of two prime numbers will be 1.

Unity is considered coprime with everything else.

The concept of coprime is meaningful only with natural numbers.

Two successive integers are always coprime. For eg: (2,3), (11,12), (99,100) and so on; they have 1 as their HCF.

The sum of any two coprime numbers is always coprime with the product of the same two coprime numbers. For eg: The sum of 2 and 3 is 5 and the product of 2 and 3 is 6. Hence, 5 and 6 are coprime numbers.

If p and q are coprime numbers then p2 and q2 are also coprime numbers. For example, the squares of 5 and 6 i.e. 25 and 36 are also coprime to each other.

Some more examples with solutions

The concept of coprime numbers can also be extended to polynomial expressions.

Example 1

x^{2}−4𝑥+3 is coprime with x^{2}−6𝑥+8

Solution

Factors of x^{2}−4𝑥+3 = (x-1)(x-3)

Factors of x^{2}−6𝑥+8 = (x-4)(x-2)

Therefore none of the factors is common between the two polynomial expressions. So they are coprime.

Coprime

Example 2

x^{2}−4𝑥+3 is not coprime with x^{2}-3x+2

Solution

Factors of x^{2}−4𝑥+3 = (x-1)(x-3)

Factors of x^{2}-3x+2 = (x-1)(x-2)

Therefore, (x-1) is a common factor between the two polynomial expressions. So they are not coprime.

Not Coprime

Conclusion

This blog tells us about Coprime numbers, properties and examples which help us find out whether a number is coprime or not. The math journey around co-prime numbers starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds.

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Frequently Asked Questions on Coprime Numbers

Out of 26, 13, and 34, which numbers are coprime?

Factors of 13: 1,13

Factors of 26: 1, 2, 13, 26

Factors of 34: 1, 2, 17, 34

The pair of (13, 26) and (26, 34) are not co primes. But a pair of (13, 34) is a co-prime as there are no factors in common except 1. As a whole, the pair is not a coprime pair.

What is the LCM of coprime numbers?

The LCM of the taken pair of coprime is equal to the product of the two numbers. For eg: 10 and 21 are coprime. The LCM of 10 and 21 will be 210 i.e. (21*10).

Which is the only pair of 3 coprime numbers?

The pair of (1, 2, 3) is the only continuous pair of 3 coprime numbers to each other. If a pair of 3 numbers contains 2 odd numbers, they can never be coprime. But if the pair of 3 numbers contain odd numbers, they can be coprime. We will have to check, in the way as shown in the examples above.

If two numbers are coprime then one of them is always divisible by the other?

No. If a number is divisible by the other number of the pair, then the pair is not a coprime pair.

Are 30 and 415 are coprime numbers?

Factors of 30 = 1, 2, 3, 5, 10, 15, 30

Factors of 415 = 1, 5, 83, 415

On comparing the factors of these two numbers 30 and 415, we see that both these numbers have factors in common. The HCF for this pair is 5. Therefore the pair is not coprime.

Are 15 and 37 are coprime numbers?

Factors of 15 = 1, 3, 5, 15

Factors of 37 = 1, 37

On comparing the factors of these two numbers 37 and 15, we see that both these numbers have no factors in common. The HCF for this pair is 1. Therefore the pair is coprime.

Are 216 and 215 coprime numbers?

Two numbers that are successive integers are always coprime. Hence the numbers are coprime numbers.

External references

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