Show there is no positive integral n we with phin15 Show the

Solution

Let us assume there exists a positive integer with Phi(n) = 15

We also know that n = p1^(a1) p2^(a2) ... pt^(at)

Phi(n) = p1^(a1-1) (p-1)... pt^(at-1)(pt-1)

So no prime p > 16 divides n, otherwise phi(n) > (n-1) > 15

Hence the only possible prime factors are 2,3,5,7,11,13

But the numbers 7,11 and 13 can be eliminated, so we can write n = 3^a * 5^b, then phi(n) won\'t be divisible by 8, hence the phi(n) = 15 will not have any integral solution