For a positive integer n, \phi(n) denotes the number of positive integers less than or equal to $n$ that are relatively prime to $n$. What is $\phi(20)$?

RedDragonl Oct 6, 2024

#1**+1 **

There's many ways to do this problem, but I prefer using Euler's Totient Function, mainly because when facing larger and more complicated numbers, listing is not going to do the trick.

So here's the Totient Function. It essentially states that

\(\frac{\text{number we start with}}{\text{product of the prime factors}} \cdot \text{product of every prime factor -1}\) and this is our final answer

Thus, we can apply this to 20.

We have \(20 = 2^2 * 5\)

2 and 5 are the two prime factors, so we have

\(\frac{20}{2*5} \cdot (2-1)(5-1) = 2 * 4 = 8\)

Thus, 8 is the final answer.

If you want to learn more, here's another question that revolves around the equation.

https://web2.0calc.com/questions/euler-s-totient-function-problems

Thanks! :)

NotThatSmart Oct 7, 2024