Give a one-sentence synopsis of the proof of Theorem 20.8.

Data from 20.8 Theorem (Euler's Theorem) If a is an integer relatively prime to n, then a^φ(n) - 1 is divisible by n, that is, a^φ(n) = 1 (mod n). 

Proof If a is relatively prime ton, then the coset a + nZ of nZ containing a contains an integer b < n and relatively prime to n. Using the fact that multiplication of these cosets by multiplication modulo n of representatives is well-defined, we have  a^φ(n) = b^φ(n) (mod n). 

But by Theorems 19.3 and 20.6, b can be viewed as an element of the multiplicative group G_n of order φ(n)elements of Z_n relatively prime to n. b^φ(n) = 1 (mod n),  and our theorem follows.