Euler's Φ Function Proofs

hi Quote:

Prove or Disprove

1) With two exceptions, if 1 ≤ n < m then Φ(n) <Φ(m)

Rules I used:

1)if p is prime, then Φ(p)= p-1

2) if p is prime, then for all e > 0, Φ(p^e) = p^e-1 * (p-1)

Attempt for problem 1).

Statement is true with the exceptions of cases b and c.

I considered 4 cases.

a)When n and m are both primes.

Φ(n) < Φ(m) is true by induction

b) When n and m both are not primes.

Φ(n) <Φ(m) is false by a counter example

Φ(n=25) < Φ(m=26) is false

c) n is prime and m is not prime.

False by counter example n=13 and m=14

d) n is not prime and m is prime.

True but I do not know a formal way to prove this.

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Prove or disprove:

2) There is no n with Φ(n)= 14.

Guess: Statement is true.

Case 1: If n is a prime then Φ(n)= n-1

n can only be 15, but 15 is not a prime.

therefore false.

Case 2: If n is not a prime.

There exist a factorization of primes. so rule 2 can be applied.

2) if p is prime, then for all e > 0, Φ(p^e) = p^e-1 * (p-1)

14 has two factorization 1*14 and 2*7

When p = 2

14= 2^(e-1) * (2-1)

when p = 7

14 = 7^e-1 * ( 7-1)

For both of these equations there are no such integer e which can satisfy these equation.

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3)For every m ≥ 3, no matter how large n might be, Φ(m) has to be an even number.

True. I dont know how to prove this.

I need help with all three problem. my proof are informal.

Please comment and offer any advice that would help me.