Help with abstract algebra please?

Question

Find all values of n that solve the equation φ(n) = n/2 using the following formula:

φ(n) = n(1 - (1/p_1))(1 - (1/p_2)*...*(1-(1/p_r)) <- where p_1,p_2,...p_r are the distinct primes that divide n.

Answer 1

Let's take the primes in ascending order, p_1 < ... < p_r.
If n = p_1^k_1 . p_2^k_2 ... . p_r^k_r, then
n(1 - (1/p_1))(1 - (1/p_2)*...*(1-(1/p_r))
= p_1^(k_1-1)p_2^(k_2-1) ... p_r^(k_r-1) . (p_1-1) (p_2-1) ... (p_r-1)
(For this to be equal to n/2, we must have p_1 = 2.)
If r > 1, then p_r^k_r divides n/2 since p_r is coprime with p_1 = 2. But (p_1-1), ..., (p_r-1) are all less than p_r, so p_r^k_r does not divide p_1^(k_1-1)p_2^(k_2-1) ... p_r^(k_r-1) . (p_1-1) (p_2-1) ... (p_r-1).
So we must have r = 1, i.e. n = 2^k
φ(n) = 2^k(1 - 1/2) = 2^k (1/2) = n/2.

So n solves the equation φ(n) = n/2 <=> n = 2^k for some positive integer k.