Basic Number Theory

Linear Congruences

In general, then, to solve ax b (mod m) we need to find the inverse of a modulo m, multiply and simplify:

ax b (mod m)

aax ab (mod m)

x ab (mod m)

The difficult part is finding the inverse a. If aa 1 (mod m) then 1 − aa = km for some integer k. This means, however, that

If gcd(a,m) = 1 then we can "run the Euclidean algorithm backwards" to express 1 as a linear combination of a and m; this is exactly what we need to do to find a.