Types of Numbers- Co-Prime Numbers

What are the Co-prime numbers?

Co-prime numbers are also known as relative prime numbers or mutually prime numbers. It is defined as the numbers which have 1 as their greatest common divisor or highest common factor. It is the prime number that divides only one number and cannot divide the other number. For example; 7 and 8, the factors of 7 is 1×7 while that of 8 is 2×2×2×1; so the highest common factor between 7 and 8 is 1 so 7 and 8 are co-prime numbers.

2 and 4 are not co-prime numbers because their common factor is 2 as the factor of 2 is 2×1 and that of 4 is 2×2×1.

Co-prime numbers definition:

Co-prime numbers can be defined as the set of two or more positive integers which has only 1 as their highest common factor or greatest common divisor. For example; 2,5 and 7 have 1 as their greatest common factor as the factors of 2 are 2×1, 5=5×1, 7=7×1 so their common factor is 1.

Co-prime polynomials are the polynomials algebraic equations whose highest common factor is 1.

List of Co-prime numbers:

Some of the pairs of co-prime numbers are:

(1,2); (2,3); (1,51); (3,5); (13,17); (1,3); (2,5); (1,52); (3,7); (13,19); (1,4); (2,7); (1,53); (3,11); (13,23); (1,5); (2,9); (1,54); (3,13); (13,29); (1,6); (2,11); (1,55); (3,17); (13,31); (1,7); (2,13); (1,56); (3,19); (13,37); (1,8); (2,15); (1,57); (3,23); (13,41); (1,9); (2,17); (1,58); (3,29); (13,43); (1,10); (2,19); (1,59); (3,31); (13,47); (1,11); (2,21); (1,60); (3,37); (13,53); (1,12); (2,23); (1,61); (3,41); (7,11); (1,13); (2,25); (1,62); (3,43); (7,13); (1,14); (2,27); (1,63); (3,47); (7,17); (1,15); (2,29); (1,64); (3,53); (7,19); (1,16); (2,31); (1,65); (3,59) and so on.

History of co-prime numbers:

Euclidean Algorithm given by Euclid developed a method for finding the highest common factor or greatest common divisor and around 300 BC, he in his book ‘Elements’ explained about this algorithm. In 1967, Joseph Stein found the greatest common divisor for negative algorithm and in 1930, another fast algorithm was founded by Derrick Henry Lehmer.

Euclid’s algorithm said that the greatest common divisor will remain the same even on subtracting the given two integers.

Properties of co-prime numbers are:

The properties of co-prime numbers are:

1. The only integers which are co-prime with every integer are -1 and 1 and they both are co-prime with 0.
2. Set wise coprimality is important for the Chinese remainder theorem in which in a set of 2 or more prime numbers the greatest common factor is 1.
3. The pair of prime numbers which are not divided by each other are the only ones called co-prime numbers.
4. All the prime numbers are always regarded as co-prime numbers.
5. The consecutive numbers are also regarded as the co-prime numbers.
6. The two odd numbers may or may not be co-prime with each other but even numbers can never refer to as co-prime numbers because their common factor will be 2.
7. If the unit number of any number or integer is 0 or 5 then it cannot be regarded as co-prime numbers; for example: (10,15); (20,45) etc. are not co-prime numbers.
8. If two numbers are co-prime with each other then their multiplication or addition will also be co-prime numbers.

How to find co-prime numbers?

There are two methods:

Method 1: 

• Check whether the number is prime or not if it is prime then it will be a co-prime number
• If the numbers are odd then maybe they are co-prime numbers and if they are even then they can never be co-prime numbers.
• If the number has 0 or 5 as the unit number then it will never be a co-prime number because it will always be divisible by 5.

Method 2: Divisibility method

• Find the factors of the integers either by the HCF method individually.
• Then, compare the factors of both the numbers or set of numbers.
• If the highest common factor is 1 then the numbers are co-prime with each other and if not, then they are non-co-prime numbers.

Application of co-prime numbers:

1. In mathematics, it is used in geometry as well as in algebraic maths.
2. It is also used mechanically in gears of the cycle so that the teeth of the gear should be co-prime with each other for uniform wear.
3. It is also applicable in pre-secure computer cryptography techniques.
4. It is used in designing machines like rotor machines etc.

Conclusion:

Co-prime numbers are the numbers or set of integers whose greatest common divisor or highest common factor is only 1. It helps in various calculations of our algebraic equations, addition or multiplications. Examples are; (1,2); (2,3) etc. Every prime number is a co-prime number but vice versa may not be true i.e. every co-prime number is not always a prime number. 1 is the integer that is co-prime with all the integers. Elucid algorithm is used to find out the greatest common divisor. These numbers are used in uniform designing of the teeth of gear of various vehicles.