Eulers totient function and exponentials

I am trying to find all integer solutions to Euler's totient function $\displaystyle \phi(3^x \cdot 5^y) = 600$

From a theorem in my book $\displaystyle \phi(n) = n(1 - 1/p1)(1- 1/p1)$

I get $\displaystyle 600 = 3^x\cdot5^y (2/3)(4/5)$

$\displaystyle 3^x\cdot5^y = 1125$

Now I am stuck because I dont know how to solve this, any suggestions?

Re: Eulers totient function and exponentials

You need to factorise 1125.

Re: Eulers totient function and exponentials

Thank you,

When I find the prime factors and their exponents x and y, am I justified in saying that these are the only ones as prime factorization in unique?

Edit- Also if anyone could show how to solve using logarithms I would be interested in knowing.

Re: Eulers totient function and exponentials

Hi dnftp! :)

Quote:

Originally Posted by

**dnftp** Thank you,

When I find the prime factors and their exponents x and y, am I justified in saying that these are the only ones as prime factorization in unique?

Yes. A prime factorization is unique.

So you get unique solutions for x and y.

Quote:

Edit- Also if anyone could show how to solve using logarithms I would be interested in knowing.

I'm afraid that normal logarithms do not apply in number theory - they yield real numbers.

As such they are not useful.

There is such a thing as a *discrete logarithm*, but let's not go there.

They are even harder to calculate than a large-number-prime-factorization.