# Thread: Eulers totient function and exponentials

1. ## 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?

2. ## Re: Eulers totient function and exponentials

You need to factorise 1125.

3. ## 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.

4. ## Re: Eulers totient function and exponentials

Hi dnftp!

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.

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.