Invertibility in Modular Arithmetic

Prove that an integer $a$ is invertible modulo $n$ if and only if $\gcd(a, n) = 1$.

In modular arithmetic, understanding when an integer has a multiplicative inverse is key to solving many problems in number theory and cryptography. The problem of proving that an integer 'a' is invertible modulo 'n' if and only if the greatest common divisor of 'a' and 'n' is 1, is a foundational concept in this area. This essentially means that for an integer to have an inverse modulo 'n', it must share no common factors with 'n', other than 1.

To prove this, one must invoke the properties of the greatest common divisor (GCD). The condition gcd(a, n) = 1 implies that a and n are coprime, meaning 'a' and 'n' do not share any prime factors. This relationship assures that there exist integers x and y, such that a * x + n * y = 1. This equation arises from the concept of linear combinations in number theory and can be proved utilizing the Euclidean algorithm or Bezout's identity. The integer x, which is determined through these methods, serves as the modular inverse of 'a' modulo 'n'.

On the other side, if 'a' is indeed invertible modulo 'n', then by definition, there exists an integer 'b' such that a * b is congruent to 1 modulo 'n'. This implies the equation a * b - k * n = 1 for some integer k. Here again, the presence of 1 as a linear combination of 'a' and 'n' supports that gcd(a, n) must be 1, confirming the coprimality condition. Such proofs and concepts form the building blocks for deeper explorations into cryptography, particularly in understanding the structure and behavior of key systems that underpin secure communications.