How can I break up a larger number into smaller numbers and exponents?

Question

I'm trying to compute the Euler's totient function for large numbers where I can't use the other rules in order to solve for the totient.

For example:

$\theta(600) = (2^3 * 3 * 5^2)$

I am confused on how you can simplify 600 to to the numbers on the right hand of the expression.

Answer 1

We have $600=2^3\cdot 3\cdot 5^2$ and $$ \phi(600)=\phi(2^3)\cdot \phi(3)\cdot\phi(5^2)=4\cdot 2\cdot 20=160. $$ The factorisation of $600$ is easy, since $100=2^2\cdot 5^2$.