Co Prime Numbers - Examples, Definition, Properties and Twin Prime Numbers

Prime numbers: Prime numbers are a set or collection of numbers that have only two factors. The factors are either 1 or the number itself. For example, 2 is a prime number because it is either divisible by 1 or 2 itself. The other examples include numbers like 3, 5, 7, 11, 13, 17, and the list goes on. The prime numbers that have a difference of 2 between them are known as twin prime numbers. 3 and 5 are twin prime as (5-3 = 2), 5 and 7, 13 and 11 are a few examples of numbers called twin prime.

Co-Prime Definition:

Co-prime numbers are different from prime numbers. Two numbers are called co-prime numbers if and only if they have 1 as their common factor. This can be demonstrated with an example:

Let us take two numbers, say 16 and 17; if we factorize each number, we get:

16 = 1 x 2 x 2 x 2 x 2, or 1 x 4 x 4, or 1 x 2 x 8

17 = 1x 17

Here the highest common multiple or HCF of the two numbers is 1; therefore, 16 and 17 are co-prime to each other.

Let us now take another example, say 18 and 24, factorizing both the numbers we find:

18 = 1 x 2 x 3 x 3

24 = 1 x 2 x 2 x 2 x 3, or 1 x 4 x 6, or 1 x 3 x 8

There are three factors common in both numbers, that is, ‘1’, ‘2’, and ‘3’. Hence, we conclude that 18 and 24 are not co-prime to each other.

Properties of Co-prime number:

Co-prime numbers contain six main properties, which are listed below-

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• 1 is the universal co-prime number. You can pair up 1 with any number, and both will be co-prime to each other.
• In a number sequence, the integer just next, and the number just previous to an integer is co-prime. For instance, let's take the number 33; therefore, the number next to 33, i.e., 34, will be co-prime to 33; similarly, the number previous to 33, i.e., 32, is also co-prime with 33.
• 0 is not co-prime with any number as it is a factor less number.
• Even numbers can never be coprime to each other because they all share a common factor other than 1, that is 2.
• If we add two co-prime numbers and we also multiply them, then the results are also co-prime to each other. For instance, 3+4 = 7, and 3 x 4 = 12, the results 7 and 12 also have only one as their common factor, and hence they are also co-prime.
• An odd and an even number are also co-prime to one another. But if the numbers have 0 and 5 at their ones or unit place, they are not co-prime as they contain an HCF = 5.
• Prime numbers are co-prime to one another.

Co-prime and Twin Prime

• Twin prime numbers are always prime numbers, while co-prime numbers can be prime numbers or composite numbers.
• The difference between any two twin prime numbers is always 2, but if we find the difference between two co-prime numbers, it can range from 1 to any number on the number line.
• Not all co-prime numbers are twin prime, but all twin prime numbers are co-prime to each other.