Here are several different ways of stating Euler's theorem. The first is how I remember it.
version 1: exponentiation mod m is well defined if the base is a unit and the exponent is an equivalence class mod ϕ(m). In other words, if [a] ∈ Zm and [b] ∈ Zϕ(m) then [a][b] = [ab] is well defined.
version 2: if [a] ∈ Zm and b ≡ bʹ mod ϕ(m) then [a]b = [a]bʹ.
version 3: [a]b + kϕ(m) = [a]
version 4: [a]ϕ(m) = 
Write the set of units of Zm. Consider what happens when you multiply all of the units by [a]. We drew a picture with an arrow from [b] to [c] if [a][b] = [c].
You proved on the homework that [a] is a unit mod m if and only if gcd(a, m) = 1.
ϕ(m) is the number of units. If m is prime, then everything except  is relatively prime to m so ϕ(m) = m − 1.
If m = pq with p and q prime then ϕ(m) = (p − 1)(q − 1). Proof: start with pq total elements of Zpq. Subtract off p multiples of q and q multiples of p. You double counted 0. Use algebra to get (p − 1)(q − 1).