Nov 30, 2022, 16:45 IST
Co-primes are those numbers that have only one common factor, namely, 1. This means that a pair of numbers is said to be co-prime when they have the highest common factor of 1. There are different ways to define co-primes. In this article, you will understand what co-prime numbers are in mathematics and how to identify common prime numbers with simple tricks and tips, along with examples.
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What are Co-prime Numbers?
Suppose that a and b are two positive integers, so they are called co-primes if and only if they have 1 as their only common factor, and thus HCF(a, b) = 1. In other words, co-primes are the set of numbers or integers with only 1 as a common factor, i.e., their highest common factor (HCF) will be 1. Co-primes are also known as relatively prime or mutually prime. There must be two numbers to form co-primes.
How to Find the Co-prime Numbers?
Consider a set of two numbers. If they have no positive integer that can divide the numbers other than 1, the pair of numbers are co-prime.
Example 1: 20 and 22
Ans. For 20 and 22:
The factors of 20 - 1, 2, 5, and 20.
The factors of 22 - 1, 2, 11, and 22.
Here 20 and 22 have only 1 common factor, which is 1. Hence, their HCF is 1, and they are co-prime.
Example 2: 20 and 25
Ans. For 20 and 25:
The factors of 20 - 1, 2,5, and 20.
The factors of 25 - 1, 5, and 25.
Here 20 and 25 have 2 common factors; they are 1 and 5. HCF is 5, and they are not a co-prime.
List of Co-Prime Numbers
The following table shows the list of co-prime numbers.
| Co prime with | Co prime numbers pairs |
| 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) |
Properties of Co-Prime Numbers
Some properties of non-primes are as follows. These are called tricks to quickly identify pairs of prime numbers.
- 1 is co-prime with every number.
- Any two prime numbers are co-primes: Since all prime number has two factors, 1 and the number itself, the common factor of two prime numbers will only be 1. For example, 4 and 7 are two prime numbers. The only common factor is 1, so they are co-prime.
- Any two consecutive numbers/integers are always prime: Take any consecutive numbers like 2, 3 or 3, 4 or 5, 6, etc.; they have 1 as their HCF.
- The sum of the two co-prime numbers is a prime number with their product: 3 and 5 are co-primes and 8 as their sum (3+5), and 15 as their product (2×3). The numbers 5 and 6 are, therefore, prime to each other.
- Two even numbers can never form a common pair because all even numbers have a common factor of 2.
- If two numbers have their unit digits 0 and 5, then they are not a co-prime numbers. For example, 10 and 25 are not coprime because their HCF is 5 (or divisible by 5).
Co-prime Numbers from 1 to 100
Various pairs of co-primes from 1 to 100 follow the above properties. Some are
(13, 14)
(28, 57)
(1, 99)
(2, 97)
(46, 67)
(75, 41), and so on.
Also, we can write any number with the combination of one as a co-prime pair, such as (22, 1), (31, 1), (4, 1), (90, 1), and (1, 100). In this way, many co-prime numbers are defined from 1 to 100.
Co-Prime Numbers and Twin Prime Numbers
We know that co-prime numbers are the numbers whose HCF is 1 (i.e.) two numbers whose common factor is only 1 are called co-prime numbers. Whereas twin prime numbers are the prime numbers whose difference is equal to 2. For e.g., the difference between 3 and 5 is 2, 3, and 5 are twin primes. The main difference between twin primes and co-primes is as follows:
The difference between two prime numbers is always equal to 2, while the difference between two prime numbers can be any number.
Twin prime numbers are always prime numbers, while co-prime numbers can also be composite numbers.
Solved Examples
Q1. Check whether 5 and 19 are co-prime.
Ans. The factors of 5 = 1, 5
Factor of 19 = 1, 19
They have 1 as their common factor. So, they are called co-prime numbers.
Q2. Check whether 25 and 40 are co-primes.
Ans. Given two numbers are: 25 and 40
25 and 40 are divisible by 5.
From the properties of co-prime numbers, 150 and 295 are not co-prime.
Alternatively,
25 = 5 × 5
40 = 5 × 2 x 2 x 2
HCF(25, 40) = 5 ≠ 1
Therefore, 150 and 295 are not co-prime.
Also Read More About- Greater Than and Less Than Symbols, Value of Pi
Frequently Asked Question (FAQs)
Q1. Is every number a co-prime number?
Ans. Every Prime Number is always co-prime to each other. As Prime Number has only two factors, one and the number itself, the common factor of two prime numbers will be 1. E.g., 7 and 11 are two Prime Numbers. Factors 7 are 1, 7, and 11 are 1, 11.
Q2. Is 0 Co-prime to any number?
Ans. 0 is not co-prime with any number as it is a factorless number. Even numbers can never be co-prime to each other because they all share a common factor other than 1, that is, 2.
Q3. What is difference between prime and Co-prime?
Ans. A prime number is a number with only one factor and the number itself. For example, 5. It has factors only 1 and 5. Co-prime numbers are a pair of numbers that have HCF 1.
Q4. What is biggest prime number ever found?
Ans. 282,589,933 − 1
Q5. Is 16 and 21 a co-prime number?
Ans. Therefore, 16 and 21 are co-prime.
