Co Prime Numbers
We have already discussed the Prime Numbers in our last article and in this article, we are discussing the Co Prime Number. As you know, Prime numbers are those numbers that have only two factors that are 1 and Itself. Co prime numbers are those numbers that have only one common factor, ie. 1. To understand co-prime numbers in much simpler terms we must have a basic idea about common factors. Common factors are those whole numbers that commonly appear as a factor of two or more numbers. For example, the common factor of 2 and 3 is 1. How?
2= 1 x 2 and 3= 1 x 3. Here, 1 appears as a factor for both the numbers so, 1 is the common factor.
A co-prime number can easily be obtained by looking at the common factors of the given numbers. If the only common factor for both the numbers appears to be 1, then and only then the given pair of numbers are co-prime. We can also say that when the highest common factor (HCF) of two numbers is 1 then the given pair of numbers are co-prime. Now, Let’s go through the article to get a better understanding of how to find out whether the given pairs of numbers are co-prime or not. You will also learn about the properties, examples, and definitions of a co-prime number.
Co Prime Number Definition
When the common factor of two given sets of numbers is 1, then the pair of numbers are co-prime numbers. Co-prime numbers are also known as Relatively Prime Numbers. Co-prime numbers always appear in pairs. You must remember that to be co-prime numbers the numbers this is not necessary for the numbers to be a prime number. Composite numbers having common factors such as 1, can also be co-prime numbers.
For example,
Example 1: 8 and 15
8=1 x 2 x 4 x 8
15= 1 x 3 x 5 x 15
Here, 8 and 15 have only 1 as a common factor and their HCF is 1. So 8 and 15 are co prime numbers.
Example 2: 3 and 5
3= 1 x 3
5= 1 x 5
Again 3 and 5 are co-prime as the only common factor of 3 and 5 is 1.
Example 2: 4 and 6
4= 1 x 2 x 4
6= 1 x 2 x 3 x 6
Now, as you can see that the common factors of 4 and 6 are 1 as well as 2, hence 4 and 6 are not co-prime numbers.
Co Prime Number Examples
As we have discussed, co-prime numbers always appear in pairs. So, there are several sets of numbers that can be coprime and have the highest common factor as 1. Let’s have a look at some Co Prime Number Examples.
| (2,9) 2=1×2 9= 1x3x9 | (3,7) 3= 1×3 7 = 1×7 | (2,47) 2= 1×2 47=1×47 |
| (1,42) 1= 1 42= 1x2x3x6x7x14x21x42 | (11,13) 11= 1×11 13= 1×13 | (14,25) 14= 1x2x7x14 25= 1x5x25 |
| (15,32) 15= 1x3x5x15 32= 1x2x4x8x16x32 | (1,10) 1= 1 10= 1x2x5x10 | (5,21) 5=1×5 21= 1x3x7x21 |
Here, these all examples have only one common factor (1) in each set. So all pairs of numbers are co-prime numbers.
How to Identify Co-prime Numbers?
Co-prime generally appears in a pair and has the highest common factor as 1. To identify whether the given set of numbers is co-prime or not, we have to follow these simple steps:
Step 1: Write all the factors of the given numbers.
Step 2: Find out all the common factors of the given numbers.
Step 3: Then Check for the highest common factor if it is 1 then the given numbers are co-prime, otherwise it is not.
For example, 6 and 9
Step 1: All the factors of 6 and 9 are
6= 1 x 2 x 3 x 6
9= 1 x 3 x 9
Step 2: All the common factors of 6 and 9= 1 x 3.
Step 3: There are two common factors for 6 and 9 i.e 1 and 3, so the given pair of numbers is not a co-prime number.
Properties of Co-prime Numbers
There are some properties of co-prime numbers that are used to identify them.
- A co-prime number pair can be formed by (1) with any number.
Eg: (1,2), (1,3), (1,8), (1,15), (1,532), (1,2568)
- Two even numbers can never be a co-prime number because they always have two common factors i.e. 1 and 2.
E.g.: (2,4), (8,10), (20,22), (18,24), (50,100). These all pairs are not coprime numbers.
- The sum of two co-prime numbers is always coprime with the product of those numbers.
E.g: (3,8); 3+8 = 11 & 3×8 = 24. So, (11,24) is also a co- prime number.
- The Highest Common Factor (HCF) of two co-prime numbers is always 1.
E.g.: (9,11). Here 9 and 11 both are co-prime numbers and their HCF is 1.
- The Least Common Multiple (LCM) of two co-prime numbers is always their product.
E.g: (9,11). Here 9 and 11 both are co-prime numbers and their LCM is 9×11= 99.
- All pairs of the two consecutive numbers are always coprime numbers and their common factor is always 1.
Eg: (2,3), (5,6), (10,11) (15,16), (16,17), (58,59), (100,101)
- Two Prime numbers are always coprime numbers because they have only one common factor i.e. 1.
E.g.: (11,13); 11=1×11 & 13= 1×13
(17,29); 17= 1×17 & 29= 1×29
(31,37); 31= 1×31 & 37= 1×37
Here, these all pairs have only one common factor i.e 1. So, these all prime numbers are co-prime numbers.
Difference between Co Prime and Twin Prime Numbers
Co-prime numbers are those sets of numbers whose highest common factor is 1 only. But, the twin prime numbers are those two sets of prime numbers whose difference is 2. I.e if we have a difference of two prime numbers as 2 then and only then it is twin prime numbers for example,
As we know that 3 and 5 is a prime number, and these two numbers are also called twin prime numbers as the difference of 5-3=2, same as 5 and 7 are twin prime numbers as their difference is also 2 (7-5=2).
| Difference between Co-prime and Twin Prime Numbers | |
| Co-prime Number | Twin Prime Numbers |
| The difference of two co-primes can be any whole number. | The difference between two twin primes is always 2. |
| Co-prime numbers can be prime or composite numbers. | Twin prime numbers are always prime numbers. |
| Co-prime numbers may or may not be twin prime numbers. | All the sets of twin prime numbers are co-prime. |
| 1 can form a coprime pair with any number. | 1 forms a twin prime pair only with 3. Ie 1 and 3 as (3-1=2) |
Important Points on Co-prime Numbers:
- The co-prime number can not necessarily be a prime number. It can be a composite number also.
- Any two given prime numbers must be a Co-prime.
- Sets of two even numbers will never be a co-prime number.
- Number 1 (one) can make a coprime pair with any number.
- Any two consecutive numbers are always a co-prime number.
Co Prime Numbers Questions
Q1: Check out the co-prime numbers from the given sets of numbers? 8 and 9; 11 and 12; 21 and 9
Answer: First of all we will find the highest common factor of all the given sets of numbers.
In 8 and 9
8= 1x 2 x 4 x 8
9= 1 x 3 x 9
So, the highest common factor is 1, hence it is a co-prime.
In 11 and 12
11= 1 x 11
12= 1 x 2 x 3 x 4 x 6 x 12
Here, the highest common factor is 1, hence it is also a Co-prime.
In 21 and 9
21= 1 x 3 x 7 x 21
9= 1 x 3 x 9
Here, we have two common factors as 1 and 3 so, the sets of 21 and 9 are not prime numbers.
Q2: What is the Difference Between Prime Numbers and Co-prime Numbers?
Answer: Prime numbers are the numbers that have two factors 1 and itself.
Whereas, Co-prime numbers are the sets of numbers having a common factor as 1 only.
Q3: Which number is the coprime of all numbers?
Answer: According to the definition and properties of co-prime numbers, There is only one number i.e 1 which can form the pair of coprime with all other numbers or integers. So, we can say that 1 is the co-prime of any number.
Q4: Check whether 250 and 148 are co primes or not.
Answer: The factors of 250 and 148 are:
250= 1 x 2 x 5 x 10 x 25 x 50 x 125 x 250.
148= 1 x 2 x 4 x 37 x 74 x 148
So, Here the common factor of 250 and 148 are 1 and 2 so, these are not coprime numbers.
We can also solve this kind of problem by looking at the sets of given numbers if the given sets have both the even numbers then it will never be co-prime numbers.
Q5: Check whether 40 and 78 are co-primes or not.
Answer: This can be solved directly as 40 and 78 are both even numbers then 40 and 78 are not coprime numbers.
Q6: If a and b are two co-prime numbers, then the value of a3 and b3 can be coprime or not.
Answer: Yes, the value of a3 and b3 is also prime number.
If we take a= 5 and b=7 then
a3 = 53 =125
b3 = 73 = 343,
125 and 343 are also a co-prime, as
The factors of 125 and 343 are:
125= 1 x 5 x 25 x 125.
343= 1 x 7 x 49 x 343.
Having the highest common factor as 1, hence co-prime numbers.
Q7: From given numbers 23, 25, 12 and 18, find out all the pairs of co-prime?
Answer: Two co-prime numbers have the highest common factor as 1.
So, The factorization of all the numbers are:
23= 1 x 23
25= 1 x 5 x 25
12= 1 x 2 x 3 x 4 x 6 x 12
18= 1 x 2 x 3 x 6 x 9 x 18.
So, from factorization there are 5-sets of co-prime, they are:
(23 , 25) (23 , 12) (23 , 18) (25 , 12) (25 , 18)
Q8: Find out the highest common factors (HCF) for the following numbers using factorization methods?
- 32 and 43
- 28 and 64
Answer: The factors of all the given numbers are:
- 32 and 43
32= 1 x 2 x 4 x 8 x 16 x 32 x 64
43= 1 x 43
So, the HCF Is 1
- 28 and 64
28= 1 x 2 x 4 x 7 x 14 x 28.
64= 1 x 2 x 4 x 8 x 16 x 32 x 64
Here, common factors are 1, 2, and 4
So, The HCF is 4.
Q9: Find out the largest number that can divide both 225 and 340?
Answer: The largest numbers that divide both 225 and 340 is the number that is the highest common factor for 225 and 340.
So, by factorization of 225 and 340, we get
225= 1 x 3 x 5 x 9 x 15 x 25 x 45 x 75 x 225.
340= 1 x 2 x 4 x 5 x 10 x 17 x 20 x 34 x 68 x 85 x 170 x 340
Now, the common factors are: 1, and 5
Hence, 5 is the largest number that can divide both 225 and 340.
Q10: Show that 61 and 67 are co-prime numbers?
Answer: Co-prime numbers are those numbers that have only a common factor which is 1.
So, the factorization of 61 and 67 are
61= 1 x 61
67= 1 x 67
So, (61, 67) are co-prime numbers.
We also know that a set of two prime numbers are always coprime numbers.