Solving Modular Linear Equations

ax $\equiv$ b (mod n)

Given a, b, n > 0; find x.

Let d = gcd(a, n)

Solvable iff d $\mid$ b

Theorem 33.23 If d $\mid$ b and d = ax' + ny' (as computed by Extended-Euclid) then one solution is x₀ = x'(b/d) mod n.

Theorem 33.24 Given one solution x₀, there are exactly d distinct solutions, modulo n, given by for i = 0, 1, 2, $\ldots$, d-1.