Help with abstract algebra please?

Question

Find all the values of n that solve the following equation
φ(n) = n/6

by using the formula
φ(n) = n(1 - (1/p_1))(1-(1/p_2))...(1-(1/p_r))

Answer 1

I will show that there are no values of n that solve
φ(n) = n/6.
First, note that n must be a multiple of 6.
Second, if you manipulate the formula for
φ(n), and decompose n into its prime factors
i.e., n = p_1^a_1 ... p_r^a_r,
you can rewrite it as
φ(n) = p_1^(a_1-1) (p_1-1) ... p_r(a^r-1)(p_r-1).
This says lower the exponent of each prime factor p_i
of n by 1 and multiply by p_i-1. Do this for all i, i = 1,r.
Example: Let's compute φ(35) = φ(5*7)
It is 4*6 = 24.
In our case let
n = 2^i 3^j p_1^a_1 ---p_t^a_t,
where the p_i are distinct primes greater than 3.
Then
φ(n) = 2^(i-1)*3^(j-1)*2 * p_1^(a_1-1)(p_1-1)... p_t^(a^t-1)*(p_t-1).
The extra 2 comes from 3-1.
This gives
φ(n) = 2^i*3^(j-1)...p_t^a_t*(p_t-1).
So
6φ(n) = 2^(i+1)*3^j ...p_t^a_t(p_t-1).
But this has one more 2 in it than n, so
6φ(n) can never equal n.
As an aside, note that if we replace the 6 by 3 we get
φ(3*2^k) = 2^k
so n = 3*2^k solves 3φ(n) = n for all k.
Hope that helps!