Coprime Numbers

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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.

What are coprime numbers? Showing 6 and 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.

What are coprime numbers? Showing 5 and 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
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 to remember
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.

Example of coprime numbers; find HCF of 161 and 192

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 \((13,5)=65\).

Practice Questions

 
 
 
 
 
 
 
 
 
 
 
 

Frequently Asked Questions (FAQs)

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.