How to break down the math symbols in this equation

Question

$$\frac{n}{\phi(n)}=\frac{n}{n\prod_{p|n}\left(1-\frac{1}{p}\right)}=\frac{1}{\prod_{p|n}\left(1-\frac{1}{p}\right)}$$

How do I learn to understand these equations by myself as I can't seem to find the mathematical notation descriptions online?

Answer 1

The big pi, $\prod$ denotes a product. The subscript on this tells you which numbers this product is over. In this example, the subscript says $p|n$ which means "$p$ divides $n$" i.e. the product is over all the prime numbers $p$ that divide $n$ (the prime factors of $n$). $\phi(n)$ denotes the Euler-Totient function. This counts the number of integers $m<n$ which are co-prime to $n$, i.e. have $\gcd(m,n)=1$.

As an example, say we have $n=105=3\times5\times7$. Then $$\prod_{p|n}\left(1-\frac1p\right)=\left(1-\frac13\right)\times\left(1-\frac15\right)\times\left(1-\frac17\right)=\frac{16}{35}$$