In this mini-lesson, we will explore the world of coprime numbers by finding the answers to the questions like how to find coprimes, how the coprimes are determined, and how to find the HCF of 2 numbers while discovering the interesting facts around them.
Two numbers are co-prime if their common factor is only 1.
In this simulation given below, enter any 2 numbers in the boxes. Click the "Get factors" button, get the factors of the numbers entered, and check if they are coprime.
Doesn't it sound interesting! Let's explore further.
Lesson Plan
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What are Coprime Numbers?
If the only common 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 GCF.
If their GCF is \(1\), we can say that they are coprime.
Example 1:
Let us consider two numbers \(5\) and \(9\)
The factors of \(5\) are \(1\) and \(8\).
The factors of \(9\) are \(1,3\) and \(9\).
The factor that is common to both \(5\) and \(9\) is \(1\).
GCF of \((5,9)=1\).
Thus, \((5, 9)\) is a coprime pair.
Example 2:
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 GCF \((6, 10) = 2\)
Thus, \((6,10)\) is NOT a coprime pair.
Coprime numbers are also referred to as relatively prime or mutually prime numbers.
Unlock the Mystical World of Coprime Numbers
Is it not fun to learn about the coprime numbers? Would you be interested in the other lessons around this concept?
| Click on a mini-lesson to explore further! |
| Factors |
| Composite Numbers |
| Least Common Multiple |
| Prime Factorization |
| Prime Number |
| Perfect Numbers |
- Any two prime numbers are coprime.
- Any two consecutive numbers are coprime.
- \(1\) forms a coprime pair with any other number.
- A prime number is coprime with any other number that is not it's multiple.
- Two even numbers are NEVER coprime.
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, GCF \((5, 9)=1\)
- 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 \(5 \times 9=45\)
Coprime Numbers From 1 to 100
- 2 Prime numbers have only 1 as their common factor. Consider 29 and 31.
29 has 2 prime factors. 1 and 29 only
31 has 2 prime factors. 1 and 31 only.
29 and 31 are prime numbers. They have only one common factor 1.
Thus they are co-prime.
We can check any two prime numbers and get them as coprime.
For example, 2 and 3, 5 and 7, 11 and 13, and so on.
- Any 2 consecutive numbers have 1 as their common factor.
Consider a few pairs of such numbers. Let us try with 14 and 15.
| Numbers | 14 | 15 |
| Factors | 1,2,7,14 | 1,3,5,15 |
|---|---|---|
| Common Factor | 1 | |
There are multiple such combinations as 1 is the only common factor.
- Two numbers are coprime if their GCF is \(1\) and vice versa.
- 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 \((13,5)=65\). |
Interactive Questions
Let's Summarize
The mini-lesson targeted the fascinating concept of coprime numbers. 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. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.
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Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.
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.