Coprime and pairwise coprime numbers
Recall that coprime or relatively prime or mutually prime integers are the integers that have no common divisors other than ± 1. The set of integers is pairwise coprime if a and b are coprime for every pair (a, b) of different integers in the set.
The property of pairwise coprime is stronger than the property of mutual prime - pairwise coprime numbers will also be mutually prime, but the opposite is not true. Obviously, for just two integers, the concepts "coprime" and "pairwise coprime" coincide.
The rule for checking for mutual prime integers follows from the definition of coprime numbers - if the GCD (greatest common divisor) of several numbers is 1, then these numbers are coprime.
To check if numbers are pairwise coprime, you can use the following property: for pairwise coprime numbers, the LCM (least common multiple) is equal to the absolute value of their products. That is, it is enough to find the LCM of several numbers and compare them with their modulo product. If they are equal, the numbers are pairwise coprime.
For example, 126 435 277 are coprime, but not pairwise coprime. But 127 435 277 are pairwise coprime.
You can read about the method of calculating GCD and LCM of several numbers here.
- • The greatest common divisor and the least common multiple of two integers
- • The Greatest Common Divisor (GCD) and the Least Common Multiple (LCM) of several numbers
- • The greatest common divisor of two integers
- • The Greatest Common Factor of several numbers
- • Least Common Denominator for several fractions