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Co-Prime Numbers are a set of numbers that have no common divisor/ factor other than 1. This means that the HCF or greatest common divisor of these numbers is 1.
- Co-Prime Numbers are known as Relatively Prime Numbers.
- If the greatest common divisor of the two numbers a and b is 1, then a and b are relatively prime numbers.
- In this case, (a, b) are called co-prime pairs.
- It requires a minimum set of two numbers to form co-prime numbers.
- It would help if you remembered that it is not necessary for these numbers to be prime numbers.
- Composite numbers having common factors equivalent to one are also considered co-prime numbers.
- 8 and 15 belong to the set of these types of numbers.
Key Terms: Coprime Numbers. HCF, Prime Numbers, Natural Numbers, Divisor, Dividend, Composite Numbers, Common Factors, Numbers, Even Numbers, Odd Numbers, Twin Prime Numbers
What are Co-prime Numbers?
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Co-prime numbers are a set of numbers that do not have a factor other than one. It means their highest common factor is one. These numbers can easily be determined by looking at the common factors of the given numbers.
- They are also known as mutually prime numbers.
- In co-prime pairs, at least two numbers are required to form a set of relatively prime numbers.
- Suppose that x and y are two positive integers; the greatest common divisor is 1.
- They are relatively prime only if HCF (x, y) = 1.
- That is, a co-prime is a set of numbers or integers.
Example of What are Co-prime Numbers?Example: For example, (4 and 5), (5, 7, 9, 4) have only 1 as their highest common factor. |
Co-Prime Numbers
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How to find Co-prime Numbers
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In a set of two numbers with no positive integer that divides only with one and no other number, then the pair of numbers is co-prime. These numbers appear in pairs.
- First, write the factors of all the given numbers.
- Determine the common factors of the given numbers.
- Next, determine the highest common factor of these numbers.
- If the highest common factor is one, then numbers are considered prime numbers; otherwise, it is not.
Example of How to find Co-Prime NumbersExample 1: For Example 4 and 5 For 4 and 5: The factors of 4 are 2, 2, and 1. The factors of 5 are 5, and 1. Here 4 and 5 have only one common factor that is 1. Hence, their HCF is 1 and is co-prime. Example 2: For Example 18 and 35 For 18 and 35: The factors of 18 are 1, 2, 3, 6, 9, and 18. The factors of 35 are 1, 5, 7, 35. Here 18 and 35 have only one common factor that is 1. Hence, their HCF is 1 and is co-prime. Example: For Example 9 and 18 For 9 and 18 : The factors of 9 are 1, 3, and 3. The factors of 18 are 1, 3, 3, and 2. Here 9 and 18 have 1 and 3 as a common prime factor. Hence, their HCF is 3 and is not a co-prime. |
Properties of Co-Prime Numbers
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The co-prime numbers can be easily identified using some of the properties described below.
- A co-prime common with every number is 1.
- One is always the greatest common divisor (HCF) of two relatively prime numbers.
- Two even numbers have two as the common factor and cannot be co-prime numbers.
- This case is found in (2,4) and (4,6).
- The product of two co-prime numbers is coprime with the sum of two co-prime numbers.
- It can be explained with the help of 4 and 5, which are co-prime numbers.
- Here, 5 + 4 = 9 is coprime with 4 × 5 = 20.
Prime numbers are always co-prime to each other. Every prime number is divisible by 1, and the number itself thus has only two factors. The only common factor of two prime numbers will be 1.
- The two consecutive pairs of numbers are also Co-prime numbers.
- All two consecutive pairs of numbers have one as their common factor.
- If two numbers have 0 and 5 at their unit's place, then they are not co-prime to each other.
Properties of Co-Prime NumbersExample 1: For example, 7 and 3 are two prime numbers. Factors of 7 are 1, 7, and factors of 3 are 1, 3. The only common factor is 1, and therefore they are co-prime. (29 and 31), (2 and 3), (17 and 19) all these are prime numbers as well as co-prime numbers. Example 2: For example, 20 and 25 are not Co- prime. Their HCF is 5, and they are divisible by 5. Example 3: For example, (2 and 3), (4 and 5), (11 and 12) all are consecutive numbers as well as co-prime numbers. |
Co-Prime Numbers List
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There are infinite pair of co prime numbers from 1 to 100. Some of the numbers from are listed below:
| Co prime with | Co prime numbers range |
|---|---|
| 1 | (1, 2), (1, 3), (1, 4), (1, 5) (1, 6),….., (1, 20),…. |
| 2 | (2, 3), (2, 5), (2, 7), (2, 9), …, (2, 15),….. |
| 3 | (3, 4), (3, 5), (3, 7), (3, 10), (3, 11),…., (3, 20),… |
| 4 | (4, 5), (4, 7), (4, 9), (4, 11), (4, 13), (4, 15),…. |
| 5 | (5, 6), (5, 7), (5, 8), (5, 9), (5, 11), (5, 12),… |
Co-Prime Numbers and Twin Prime Numbers
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The relatively prime numbers are numbers with an HCF of 1, that is, two numbers with a greatest common divisor of 1 are called Co-prime prime numbers. Twin prime numbers, on the other hand, are primes whose difference is always equal to 2.
- The main difference between twin primes and Co-prime numbers is 2.
- The difference between two twin primes is always equal to 2, but the difference between two co-prime numbers can be any number.
- Twin prime are always prime, but relatively prime numbers can also be composites.
Example of Co-Prime Numbers and Twin Prime NumbersExample: For example, the difference between 7 and 9 is 2, so 7 and 9 are twin primes. |
List of Prime Numbers and Twin Prime Numbers
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Following is a list of Twin Prime numbers between 1 to 1000:
| Numbers range | Twin prime numbers |
|---|---|
| 1 to 50 | {3, 5}, {5, 7}, {11, 13}, {17, 19}, {29, 31}, {41, 43} |
| 51 to 100 | {59, 61}, {71, 73} |
| 101 to 200 | {101, 103}, {107, 109}, {137, 139}, {149, 151}, {179, 181}, {191, 193}, {197, 199} |
| 201 to 300 | {227, 229}, {239, 241}, {269, 271}, {281, 283} |
| 301 to 400 | {311, 313}, {347, 349} |
| 401 to 500 | {419, 421}, {431, 433}, {461, 463} |
| 501 to 1000 | {521, 523}, {569, 571}, {599, 601}, {617, 619}, {641, 643}, {659, 661}, {809, 811}, {821, 823}, {827, 829}, {857, 859}, {881, 883} |
The list of Prime numbers between 1-100 is as follows:
| Prime Number List |
|---|
| 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 |
Things to Remember
- Co-Prime numbers belong to the category of numbers whose HCF is equal to one.
- The product of any two of these is always co-prime with their sum.
- There are almost 25 prime numbers between 1-100
- Two even numbers cannot be co-prime numbers
- The difference between two twin primes is always equal to 2
- Prime numbers are always co-prime to each other
- At least two numbers are required to form a set of Co-prime numbers.
- If two numbers have 0 and 5 at their unit's place, then they are not Coprime to each other.
- The two consecutive pairs of numbers are also Co-prime numbers.
Also Read:
Sample Questions
Ques. Find out whether 11 and 31 are co-prime. (2 Marks)
Ans. 11 and 31 are two prime numbers; thus, they are co-prime to each other.
The factors of 11 are 1, 11 and the factors of 31 are 1, 31.
They have only 1 as their common factor. So, they are co-prime numbers.
Ques. Find out whether 11 and 12 are co-primes. (2 Marks)
Ans. Given two numbers are 11 and 12 consecutive numbers and thus are co-prime numbers 150 and The factors of 11 are 1, 11 and the factors of 12 are 2, 2, 3, 1.
They have only 1 as their common factor. So, they are co-prime numbers
Ques. Find out whether 15 and 20 are co-primes. (3 Marks)
Ans. Given two number are: 15 and 20
15 and 20 are divisible by 5.
15 = 3 × 5
20 = 5 × 4
HCF(15, 20) = 5 ≠ 1
Therefore they have two common factors 5 and 1 and hence they are not coprime numbers.
From the properties of coprime numbers,
If two numbers have 0 and 5 at their unit's place, then they are not Coprime to each other. We can say 15 and 20 are not coprime.
Ques. Find out whether 17 and 168 are co-prime numbers. (2 Marks)
Ans. Given two number are: 17 and 168
Factors of 17 = 1, 17
Factors of 68 = 1, 2, 4, 17, 34, 86
Common factors of 17 and 68 and 1, 17. Since they have two common factors they are not coprime numbers.
Ques. Find out whether 30 and 215 are co-prime numbers. (2 Marks)
Ans. Given two number are: 30 and 215
Factors of 30 = 1, 2, 3, 5, 6, 15, 30
Factors of 215 = 1, 5, 43, 215
Common factors of 30 and 215 = 1 and 5. Since they have two common factors they are not coprime numbers.
Ques. What is the Difference Between Twin prime Numbers and Co-Prime Numbers? (2 Marks)
Ans. A Twin Prime Number is defined as a Number whose difference is always equal to 2. Co-Prime Numbers have a Common factor as 1 but their difference is not always one (1).
Ques. What are 3 pairs of co prime numbers from 80 to 100. (1 Marks)
Ans. The 3 pairs of co prime numbers from 80 to 100 are 87,88 91,92 and 99,100.
Ques. Is 1 Coprime number to all numbers? (1 Mark)
Ans. Yes, the HCF of 1 and any number is 1 itself. Hence, by the definition of co-prime numbers, 1 is said to be coprime with all numbers.
Ques. How to find coprime numbers? (2 Marks)
Ans. To find the coprime of a number,the first step is to identify the number (if both are even , then they cannot be co-prime numbers) and then find the factors of the number. Then, choose any number and find the common factor number in both. All the numbers which do not have any common factor other than 1 will be the co-prime number.
Ques. Can a single number be a Co-prime Number? (2 Marks)
Ans. No, a single number cannot be a co-prime number as they are always in pairs.( At least two numbers together can form a co-prime number).
Ques. If 15 and 19 are co-prime, what would be their HCF? (1 Mark)
Ans. It is given that 15 and 19 are co-prime. They cannot have any common factor other than 1. Hence, their HCF is 1.
Ques. What is the difference between co-prime and twin prime numbers? (3 marks)
Ans. The difference between co-prime and twin prime numbers are as follows:
| Co-prime Numbers | Twin Prime Numbers |
|---|---|
| The relatively prime numbers are numbers with an HCF of 1 are co-prime numbers. | Twin primes numbers, on the other hand, are primes whose difference is always equal to 2. |
| They can be prime or composite numbers. | They are always prime numbers. |
| In this one can form pair with any number. | In this 1 forms a twin prime pair only with 3. |
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