For a positive integer N, let φ(N) denote the number of positive integers (including unity) which are less than N and coprime to it. Which of the following statements are true?
a. If N is not equal to M then φ(NM) = φ(N)φ(M).
b. If N > 2, then φ(N) is always even
c. If p is a prime and if N = p^k , k ∈ N, then φ(N) = N(1-1/p)
i think all option a) and C) are correct according to euler theorem..
But still im doubting about my answer. If anybody body help me i would be very thankful to him..
a) is true only if $N$ and $M$ are coprime - i.e. $\phi$ is multiplicative, but not completely multiplicative
b) Edit: True!
c) True! $\phi(p) = p-1$ for prime $p$, and there are $p^{k-1}$ integers, $n$ such that $1 \leq n leq p^{k}$ which are divisible by $p$. Combine these facts :)
EDIT: to prove a) for coprime $N$ and $M$, consider the sets $A,B,C$ which are less than and coprime to $N$, $M$ and $NM$ resp. Then by the Chinese Remainder Theorem there is a bijection between $A \times B$ and $C$
