number theory and abstract algebra

Just hoping for some help with my practice exam. Any help on any of the questions would be greatly appreciated as my lecturer is absolutely shocking. Cheers.

(2)b

Let n ∈ N and let p be a prime. Show that if p | n then φ(np) = pφ(n).

Hint: consider the prime factorisation of n.

(17)

Show that the inverse of 5 modulo 101 is 5^99.

Use repeated squaring to simplify 5^99 (mod101)

Hence solve the equation 5x = 31(mod101)

(31)

Let P(R) be the set of all subsets of R; that is, P(R) = {X | X ⊆ R}. Let F be the set of all functions R → R. Define R : F → P(R) by R(f ) = {x ∈ R | f (x) = 0}. Prove

that R is surjective but not injective.

(36)b

In which of the following is (G, ∗) a semigroup? In which is it a group? Prove your answers.

G = N, a ∗ b = max{a, b}.

G = R \ { 1/2 }, a ∗ b = a − 2ab + b.

I know this is a lot of question but I'm really struggling and my test book doesn't have any examples. Thank you for all your time.