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For example, the integers 6, 10, 15 are coprime because 1 is the only positive integer that divides all of them. If every pair in a set of integers is coprime, then the set is said to be pairwise coprime (or pairwise relatively prime, mutually coprime or mutually relatively prime). Pairwise coprimality is a stronger condition than setwise ...
Integers in the same congruence class a ≡ b (mod n) satisfy gcd(a, n) = gcd(b, n); hence one is coprime to n if and only if the other is. Thus the notion of congruence classes modulo n that are coprime to n is well-defined. Since gcd(a, n) = 1 and gcd(b, n) = 1 implies gcd(ab, n) = 1, the set of classes coprime to n is closed under ...
The previous result says that a solution exists if and only if gcd(a, m) = 1, that is, a and m must be relatively prime (i.e. coprime). Furthermore, when this condition holds, there is exactly one solution, i.e., when it exists, a modular multiplicative inverse is unique: [ 8 ] If b and b' are both modular multiplicative inverses of a respect ...
If a is not divisible by p, that is, if a is coprime to p, then Fermat's little theorem is equivalent to the statement that a p − 1 − 1 is an integer multiple of p, or in symbols: [1] [2] (). For example, if a = 2 and p = 7, then 2 6 = 64, and 64 − 1 = 63 = 7 × 9 is a multiple of 7.
The natural numbers m and n must be coprime, since any common factor could be factored out of m and n to make g greater. Thus, any other number c that divides both a and b must also divide g . The greatest common divisor g of a and b is the unique (positive) common divisor of a and b that is divisible by any other common divisor c .
An element a of Z/nZ has a multiplicative inverse (that is, it is a unit) if it is coprime to n. In particular, if n is prime, a has a multiplicative inverse if it is not zero (modulo n). Thus Z/nZ is a field if and only if n is prime. Bézout's identity asserts that a and n are coprime if and only if there exist integers s and t such that
The converse of Euler's theorem is also true: if the above congruence is true, then and must be coprime. The theorem is further generalized by some of Carmichael's theorems . The theorem may be used to easily reduce large powers modulo n {\displaystyle n} .
Since every rational number has a unique representation with coprime (also termed relatively prime) and , the function is well-defined. Note that q = + 1 {\displaystyle q=+1} is the only number in N {\displaystyle \mathbb {N} } that is coprime to p = 0. {\displaystyle p=0.}