+786 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 $\phi(20)$?
+1853 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! :)
