Is there an algorithmic or methodical way to "factorise" the numbers in euler's phi function such that it becomes easily computable?
For example, $\phi(7000) = \phi(2^3 \cdot 5^3 \cdot 7)$
I'm finding it difficult to find this "factorised" version of 7000 in terms that are easily computable in the phi function.
Also, since computing $\phi$ of a prime number is easy, I thought one method would be to factorise 7000 down to multiples of ONLY prime numbers, ie:
$7000 = 7 \times 5 \times 5 \times 5 \times 2 \times 2 \times 2 $
$\phi(7000) = \phi(7) \times \phi(5) \times \phi(5) \times \phi(5) \times \phi(2) \times \phi(2) \times \phi(2)$
$\phi(7000) = 6 \times 4 \times 4 \times 4 \times 1 \times 1 \times 1 $
(Since the $\phi$ of a prime number is just that number $- 1$)
But the answer is not correct. I get $384$. Supposed to be $2400$. Why does this not work?