How to use Euler's theorem ?

Can someone please check/comment on my work?

1. Find the order of 7 modulo 172.

I did this by trial and error

7^1= 7(mod 172)

7^2= 49(mod 172)

7^3=171(mod 172) -----> 7^3 = -1(mod172)

squaring it

7^6 = (7^3)^2 = (-1)^2 = 1(mod 172)

so the order is 6

When I tried Euler method,

m =172

Φ(172) = Φ(2^2 * 43)

By rules: If p is prime then a) Φ(p)=p-1 b) Φ(p^e)=(p^(e-1))(p-1)

Φ(2^2 * 43) = 2*42 = 84

7^84= 1 (mod 172) [Can this be simplified?How?]

2. Prove that for any n, 33 divides (n^101) - n

33 has factors of 3 and 11. so n^101 - n is divisible by 11 and 3

I use fermat's theorem n^p = n (mod p)

n^p-1 = 1(mod p)

n= 1(mod 3)

n^100 = 1(mod 3)

n^100 * n= n(mod 3)

n^101 - n = 0 (mod 3)

n= 1(mod 11)

n^100 = 1(mod 11)

n^100 * n= n(mod 11)

n^101 - n = 0 (mod 11)

therefore n^101 - n divisible by 33.

Im not too sure if the proof is correct.

I would like to know how to do this problem with Euler's Theorem.

3. Calculate Φ(5525)

Φ(5525) = Φ(5^2 * 13 *17)

By rules: If p is prime then

a) Φ(p)=p-1 b) Φ(p^e)=(p^(e-1))(p-1)

Φ(5^2) = 20 Φ(13)=12 Φ(17)=16

Φ(5525) = 20*12*16 = 3840