# Coprime Numbers

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 1 What are Coprime Numbers? 2 Tips and Tricks to Find Coprime Pairs 3 Coprime Numbers From 1 to 100 4 Properties of Coprime Numbers 5 Important Notes of Coprime Numbers 6 Solved Examples of Coprime Numbers 7 Practice Questions of Coprime Numbers 8 Frequently Asked Questions (FAQs)

## What are Coprime Numbers?

If the only factor of two numbers $$a$$ and $$b$$ is $$1$$, then $$a$$ and $$b$$ are coprime numbers.

In this case, $$(a, b)$$ is said to be a coprime pair.

### How to Find Coprime Numbers?

To find whether any two numbers are coprime, we first find their HCF.

If their HCF is $$1$$, we can say that they are coprime.

Example 1:

Let us consider two numbers $$6$$ and $$10$$.

The factors of $$6$$ are $$1,2,3$$ and $$6$$.

The factors of $$10$$ are $$1,2,5$$ and $$10$$.

Factors that are common to both $$6$$ and $$10$$ are $$1$$ and $$2$$.

So HCF $$(6, 10) = 2$$.

Thus, $$(6,10)$$ is NOT a coprime pair.

Example 2:

Let us consider two numbers $$5$$ and $$9$$.

The factors of $$5$$ are $$1$$ and $$5$$.

The factors of $$9$$ are $$1,3$$ and $$9$$.

The factor that is common to both $$5$$ and $$9$$ is $$1$$.

HCF of $$(5,9)=1$$

Thus, $$(5, 9)$$ is a coprime pair.

Coprime numbers are also referred to as relatively prime or mutually prime numbers.

### Coprime Number Examples

From Example 2, we know that $$5$$ and $$9$$ are coprime numbers.

Some other coprime number examples (coprime pairs) are:

1. $$2$$ and $$3$$
2. $$15$$ and $$17$$
3. $$9$$ and $$10$$

### Coprimes Factors

The only common factor of coprime numbers is $$1$$.

Thus, coprimes factor is $$1$$.

Tips and Tricks
1. Any two prime numbers are coprime.
2. Any two consecutive numbers are coprime.
3. $$1$$ forms a coprime pair with any other number.
4. A prime number is coprime with any other number that is not its multiple.
5. Two even numbers are NEVER coprime.

## Coprime Numbers From 1 to 100

There are many coprime pairs from $$1$$ to $$100$$.

Using the above definition and "Tips and Tricks to Find Coprime Pairs," some coprime numbers from $$1$$ to $$100$$ are:

• $$13, 17$$
• $$15, 16$$
• $$1, 10$$
• $$7, 15$$

Since there are multiple such combinations, it is difficult to determine the coprime numbers list from $$1$$ to $$100$$.

## Properties of Coprime Numbers

Some properties of coprime numbers are:

• The HCF of two coprime numbers is always $$1$$.

For example:

$$5$$ and $$9$$ are coprime numbers and hence, HCF $$(5, 9)=1$$.

• The LCM of two coprime numbers is always their product.

For example:

$$5$$ and $$9$$ are coprime numbers.
Hence, LCM $$(5,9) = 45$$.

• The sum of two coprime numbers is always coprime with their product.

For example:

$$5$$ and $$9$$ are coprime numbers.
Here, $$5+9=14$$ is coprime with $$5 \times 9=45$$.

Important Notes
1. Two numbers are coprime if their HCF is $$1$$ and vice versa.
2. Coprime numbers don't need to be prime numbers.
For example, $$12$$ and $$35$$ are coprime numbers.
However, $$12$$ and $$35$$ are NOT prime numbers.

## Solved Examples

 Example 1

Show that $$161$$ and $$192$$ are coprime numbers.

Solution:

We will find the HCF of the given numbers $$161$$ and $$192$$ using the division method.

The HCF of $$161$$ and $$192$$ is $$1$$.

Thus, they are coprime numbers.

 $$\therefore$$ $$161$$ and $$192$$ are coprime.
 Example 2

If $$59$$ and $$97$$ are coprime, what would be their HCF?

Solution:

It is given that $$59$$ and $$97$$ are coprime.

They cannot have any common factor other than $$1$$.

Hence, their HCF is $$1$$.

 $$\therefore$$ HCF of $$(59, 97)=1$$.
 Example 3

If $$13$$ and $$5$$ are coprime numbers, what is their LCM?

Solution:

It is given that $$13$$ and $$5$$ are coprime numbers.

Hence, their LCM is equal to their product, $$13 \times 5 = 65$$.

 $$\therefore$$ LCM $$(13,5)=65$$.

## 1. What are coprime numbers?

If the only factor of two numbers $$a$$ and $$b$$ is $$1$$, then $$a$$ and $$b$$ are coprime numbers.

It is very difficult to determine the coprime numbers list.

## 2. What is the difference between prime and coprime numbers?

A prime number is a number that is greater than $$1$$ which has exactly two factors $$1$$ and itself.

Coprime numbers are numbers whose HCF (Highest Common Factor) is $$1$$.

## 3. How to find the co-prime of a number?

The HCF of two co-prime numbers is 1.

Thus, to find the co-prime number of a number, it is sufficient to find a number which is NOT divisible by any of the factors of the given number.

## 4. Which numbers are identified as co-prime numbers?

A single number cannot be co-prime.

Two numbers are said to be co-prime if their HCF is $$1$$.

## 5. Are 18 and 35 co-prime numbers?

The factors of $$18$$ are $$1,2,3,6,18$$.

The factors of $$35$$ are $$1,5,7,35$$.

$$18$$ and $$35$$ have no common factor other than $$1$$.

Thus, $$18$$ and $$35$$ are co-prime.

## 6. Are co-prime numbers always prime numbers?

No, co-prime numbers don't have to be prime numbers.

For example:

$$18$$ and $$25$$ are co-prime numbers as their HCF is $$1$$.

But $$18$$ and $$25$$ are NOT prime numbers.