I will denote the phi function to be &.
For example, &(36) = 36(1-1/2)(1-1/3) = 12
Find all n such that &(n) = 8.
How is this done w/o just guessing and checking? Thanks...
"so n is a multiple of 2 just since if it is multiple from another prime it will be a divisor of phi(n) is it ok for now"
this is wrong statement I forgot number theory sorry
phi(5) = 4 and 2 is not a divisor of 5
phi(15) = 2(4) = 8
there is another solutions
I can't figure exactly how I can solve it sorry
The above works for when
What if ?
. Now if we have then that and WLOG .
Thus .
WLOG if , then . This means again that . So we could have or . Or in other words or .
Thus
What if ? I'll let you do this one (hint: what's ?). The answer comes out to be .
Now what about other ? Well, it turns out that with three or more prime divisors, there are just too many factors in i.e. will all be even and distinct, thus making .
So here's all with : .