Question 1207613: 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 $\phi(5)$?
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52778)   (Show Source):
Answer by math_tutor2020(3816)  (Show Source):
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Answer: 4
Explanation
For any prime p, we have the following:
phi(p) = p-1
There are p-1 positive integers smaller than p that are relatively prime to p.
1, 2, 3, ..., p-2, p-1
This intuitively makes sense because all of these values are not factors of the prime p (well except for the trivial case 1)
Examples:
phi(7) = 6
phi(11) = 10
If we had some value not relatively prime to p, smaller than p, then it would mean p isn't prime.
For instance if 2 and p weren't relatively prime, then p = 2k and p is even. But at this point p is not prime.
Further Reading
https://mathworld.wolfram.com/TotientFunction.html
and
https://en.wikipedia.org/wiki/Euler%27s_totient_function
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