Proof
The second statement of the theorem is a straightforward consequence of the first. So, we only prove the first. There are two parts to the proof:
If
Only if
If gcd (a,n)=1, then from the
extended Euclidian algorithm it follows that there are integers
x and y such that ax + ny = 1. In
Z/nZ this translates to
a (mod n) · x (mod n)
+ 0 (mod n) ·
y (mod n)
= 1 (mod n).
a (mod n) · x (mod n) = 1 (mod n).
In particular,
If (a (mod n)) · (b (mod n)) = 1 (mod n), then there is an integer x with
So, by Section 1.2, we find gcd(a,n) = 1.