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What is Coprime?
Coprime are numbers that have no common factors. A number is coprime to another number if when you divide both by some integer, you will get a result of 1. This can be calculated using an algorithm or a math formula. If neither number has any common factors and both these numbers are prime, then these numbers are called twin primes.
- Examples of coprimes include 3 and 5, 11 and 13, 29 and 31 etc. The first few examples of twin primes are: 2 & 3, 3 & 5, 5 & 7 etc.
Coprimes are a good starting point for prime factorization, an important mathematical calculation. If two numbers are coprimes, you will get their common factors by dividing them.
- For example, if you divide 3 and 5, you will get 1 and 2 as their common factors. Since both these numbers have no common factors with 3 and 5 respectively, they are a coprime pair.
The sum of two coprimes is never divisible by any other number than one and itself. This is called Wilson’s theorem and states that if a+b is not divisible by any integer, then (4m)^2-1 for some m can be represented as a product of distinct primes which are none other than a and b.
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Examples of Coprime Numbers
1, 2, 4, 6, 8, 10 are coprime. So are 1/2 and 3/4. An easy way to check if two numbers are coprime is to just add them together. If you get a non-zero answer then they’re not coprime.
- For example -12 is not coprime with any number because 12 + (-12) = 0. So -12 would be divisible by all positive numbers (see: prime factorization).
The same goes for negative numbers.
- For example, 5 isn’t coprime with 7 because 5 + 7 = 12 which is divisible by 2 and 3. The same goes for composite numbers like 9 which isn’t coprime with 11 since 9 + 11 = 20 which is divisible by 2 and 5. The only composite number that’s coprime with itself is 1 (1 x 1 = 1).
You can also test if a number is coprime with all other numbers (perfectly coprime) by checking whether it has any factors that aren’t even.
- For example, there are no odd numbers that are perfect squares – so every odd number must be coprime with all even numbers since all even numbers have factors of 2, 4, 6, and 8 which aren’t perfect squares.
Properties Related to Coprime Numbers
On Math is Fun, we have a complete list of properties for coprime numbers. Some interesting ones include:
- Every natural number is coprime to 1, and every even natural number other than 2 is also coprime to 2.
- The product of two distinct odd primes that are not 3, 5, or 7 (such as 11 and 13) will be either a prime or a semiprime if and only if one of them has an even number of factors.
- A square can always be expressed as a sum of two or more squares (i.e., if x2 = y2 then x = y ± sqrt(y2 – x2)).
- Two nonzero numbers whose difference divides both into nonzero parts are called relatively prime to each other.
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