For what integers $n$ does $\phi(2n) = \phi(n)$? For what integers $n$ does $\phi(2n) = \phi(n)$?
Could anyone help me start this problem off? I'm new to elementary number theory and such, and I can't really get a grasp of the totient function.
I know that $$\phi(n) = n\left(1-\frac1{p_1}\right)\left(1-\frac1{p_2}\right)\cdots\left(1-\dfrac1{p_k}\right)$$ but I don't know how to apply this to the problem. I also know that $$\phi(n) = (p_1^{a_1} - p_1^{a_1-1})(p_2^{a_2} - p_2^{a_2 - 1})\cdots$$
Help

Hint: You may also prove in general that 
$$\varphi(mn)=\frac{d\varphi(m)\varphi(n)}{\varphi(d)}$$
where $d=\gcd(m,n).$
