 # Co-Prime Numbers: How to recognise and their properties

## Mathematics

Students should know what are co-prime numbers and be able to figure out the co-prime numbers from 1 to 100. Edulyte’s blog helps you learn about co-prime numbers essential for different classes, be it class 6, 9 or 12.

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## What are Co-Prime Numbers? Definition of Co-Prime Numbers

We understand a prime number as a positive integer that cannot be divided by any integer other than itself or 1.

So what are co-prime numbers, then? To understand the definition of co-prime numbers, we first understand the term. “Co” means “pair”. They are a pair of numbers whose highest common factor (HCF) or greatest common divisor (GCD) is 1. In other words, the only positive integer that is a divisor of both of them is 1. You might know these numbers by names like relatively prime or mutually prime numbers.

## List of Co-Prime Numbers

There are various co-primes number pairs from 1-100 following the specific properties. Coprime numbers are denoted as

gcd(a,b) = 1gcd(a,b) =1

or

(a,b) = 1(a,b) =1

or

as “a is prime to b”.

Based on this, you can quickly determine that there is a long list of co-prime numbers. Some of the co-prime number examples are:

(1,99), (2,3), (3,5), (5,7), (10,21), (11,13), (13,14), (17,19), (21,22), (28,57), (29,31), (41,43), (46,67), (59,61), (71,73), (75,41), (87,88), (99,100)

## How to Find Co-Prime Numbers?

Numbers are endless. So how do we find out the co-prime numbers from a given set of numbers?

To check if two given numbers are co-prime or not, you should check whether their HCF is 1.

### Example

For 11 and 23:

The factors of 11: 1 and 11.

The factors of 23: 1 and 23.

11 and 23 have only one common factor, that is 1. Therefore, the given numbers are co-prime.

Factors of 66 – 1, 2, 3, and 66

Factors of 1010 – 1, 2, 5, and 1010

The factors common to both 66 & 1010 are 11 & 22.

So, GCF (6, 10) = 2.

Thereby, (6, 10) is not the co-prime pair.

You might be quick to think that all prime integers are co-prime numbers. But if you re-consider the definition, you will realise that many non-prime numbers can also come together to make co-prime numbers.

Let us look at the example of such co-prime numbers below:

Factors of 81 = 1, 3, 9, 27, 81

Factors of 16 = 1, 2, 4, 8, 16

Common factor= 1

They are co-prime numbers

## Properties of Co-Prime Numbers

1 is co-prime with all numbers.

While looking for co-prime numbers from 1 to 100, always remember that any 2 prime numbers are co-prime to each other.

While all the prime numbers have only 2 factors, i.e., 1 and the number itself, the single common factor of 2 prime numbers will be 1. For instance, 2 & 3 are the two prime numbers. Factors of 3 are 1, 3 and factors of 2 are 1, 2. The single factor is 1, and thereby, they are co-prime.

Any 2 consecutive numbers/integers are ALWAYS co-prime. Take up any consecutive numbers like 2, 3 or 3, 4 or 5, 6 & so on. All of these have 1 as the HCF.

The sum of any 2 co-prime numbers is always co-prime with the product. For instance, 2 & 3 are co-prime and have 5 as the sum (2+3) & 6 as their product (2×3). Thereby, 5 and 6 are co-prime to each other.

2 even numbers can never form the co-prime number pair since every even number has the common factor of 2.

If out of two numbers, any one number is a prime number while the other is not a multiple of the first one, then both are co-prime. Example 11 and 21, 36 and 55.

If two numbers have the unit digits like 0 and 5, they aren't co-prime to each other. For instance, 10 & 15 aren't co-prime as the HCF is 5 (or divisible by 5).

## Co-Primes from 1 to 100

The list of Co prime numbers 1 to 100 is long. If you go through the Co prime number properties given below, you can quickly figure out whether a pair, if given numbers, show the properties of co-prime or not.

Of course, 1 is co prime with every integer between 0 to 100.

The number 2 is an exciting integer. It is co-prime with all the odd numbers.

3 has its unique property of being co-prime with all the numbers that are not themselves multiples of 3.

Prime numbers between 1 to 100 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. So a combination of any of these two numbers is a coprime pair.

As already discussed, we know that any consecutive number pair is also co-prime.

## Co-Prime and Twin Prime

Many students get confused between the two terms, Co-Prime and Twin Prime. Let us make them easy for you to understand. We know that co-prime numbers are the numbers whose HCF is 1.

Twin Prime numbers are prime numbers whose difference is always equal to 2.

75 and 41 are co-prime numbers, as their HCF is 1. But they are not twin prime numbers as when you calculate their difference; it is not 2.

On the other hand, 19 and 7 are twin prime numbers as the difference between them is 2.

There… we told you we would make it easy for you!

#### So remember

The difference between twin prime numbers should always be 2, whereas the difference between co-prime numbers can vary.

Twin prime numbers are always prime numbers, whereas co-prime numbers can also be composite numbers.

## Conclusion

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Introduction Symbols in Maths are essential as they denote different concepts and functions. Moreover, they are universally accepted and enable sharing of information symbolically. Also,

"Co" means the "pair". They are a pair of numbers whose highest common factor is 1. In other words, the only positive integer that is a  divisor of both of them is 1. You might know these numbers by names like relatively prime or mutually prime numbers.

The prime number is described as the one which has no other factor than 1 and itself. On the contrary, the co-primes are regarded in pairs & 2 numbers are considered as the co-primes in case they don’t have any other common factor than 1.

Co-prime numbers are fun to study due to their unique characteristic.

Some examples of co-prime numbers are :

1,5 2,9 5,7 1,23 1,44

1,6 2,11 5,9 1,24 1,45

1,7 2,13 5,11 1,25 1,46

1,8 2,15 5,13 1,26 1,47

1,9 2,17 5,17 1,27 1,48

1,10 2,19 5,19 1,28 1,49

1,11 2,21 5,21 1,29 1,50

1,12 2,23 5,23 1,30 7,9

LCM of the given co-prime pair is equivalent to the 2 numbers’ product. For instance, 10 & 21 are co-prime. LCM of 21 and 10 will turn out to be 210 i.e., (21×10).

Yes, when it comes to Co-prime numbers from 1 to 100, 1 is co-prime to every number. As HCF of 1 & any other number is 1 itself. Thereby, going by the definition of the co-prime number, 1 is considered as co-prime with every number.

Finding the answer to the question above barely requires any effort! All you have to determine whether they have an HCF other than 1. Suppose they don't, then they are co-prime numbers.

A shortcut method is to remember the properties of co-prime numbers. Consecutive numbers will always be co-prime numbers. 20 comes right after 19, making them consecutive numbers and hence c0-prime numbers.

For finding a number’s co-prime, firstly look for the number’s factors. Afterwards, pick a number & find the chosen number’s factors. Every number that does not have the common factor other than 1 will be co-prime of the specified number.

There are many!

• While looking for Co-prime numbers from 1 to 100? Always remember that any 2 prime numbers are co-prime to each other.
• While all the prime numbers have only 2 factors, i.e., one and the number itself, the single common factor of 2 prime numbers will be 1. For instance, 2 & 3 are the two prime numbers. Factors of 3 are 1, 3 and factors of 2 are 1, 2. The single factor is 1, and thereby, they are co-prime.
• Any 2 consecutive numbers/integers are ALWAYS co-prime. Take up any consecutive numbers like 2, 3 or 3, 4 or 5, 6 & so on. All of these have 1 as the HCF.
• The sum of any 2 co-prime numbers is always co-prime with the product. For instance, 2 & 3 are co-prime and have 5 as the sum (2+3) & 6 as their product (2×3). Thereby, 5 and 6 are co-prime to each other.
• 2 even numbers can never form the co-prime pair since every even number has the common factor of 2.
• If out of two numbers, any one number is a prime number while the other is not a multiple of the first one, then both are co-prime. Example 11 and 21, 36 and 55.
• If two numbers have the unit digits like 0 and 5, they aren't co-prime to each other. For instance, 10 & 15 aren't co-prime as the HCF is 5 (or divisible by 5).

Prime numbers are positive integers that cannot be divided by any integer other than itself or 1.

Co prime numbers are a pair of numbers whose highest common factor (HCF) or greatest common divisor (GCD) is 1.

No, it isn’t. In case the number is divisible by the pair’s other number, then that pair is not the co-prime one.

We have to calculate the HCF of 35 and 31 to determine if they are co-prime numbers.

Factors of 25: 1, 5 and 25

Factors of 3: 31 and 1.

Since their HCF is 1, that makes 25 and 31 co-prime numbers.

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