Prove ax mod n = 1 if and only if gcd(a,n) = 1

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In mod n arithmetic, when does an integer "a" have a multiplicative inverse? It will be when the modular equation ax mod n = 1 has a solution for x (so a^-1 = x). That will happen exactly when "a" and n are relatively prime (coprime). In other words, when their greatest common divisor is 1 (greatest common factor is 1). We need the Division Algorithm and the GCD is a Linear Combination Theorem to do this proof.

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Bill Kinney
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