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cwatki14
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I am trying to find all of the integers such that phi(n)=12. Clearly n=13 is one, but how do I do it for composite numbers?
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Tinyboss said:What can you say about [tex]\varphi(a)\varphi(b)[/tex] with regard to [tex]\varphi(ab)[/tex]? What about [tex]\varphi(p^n)[/tex] if p is prime?
Phi(n), also known as Euler's totient function, is a mathematical function that counts the number of positive integers less than or equal to n that are relatively prime to n.
To find all integers such that phi(n) = 12, you can use the formula phi(n) = n * (1 - 1/p1) * (1 - 1/p2) * ... * (1 - 1/pk), where n is the integer we are trying to find and p1, p2, ..., pk are the distinct prime factors of n. In this case, we can set n = 12 and solve for the prime factors p1, p2, ..., pk.
Yes, there are integers that do not have a phi(n) value of 12. For example, all even numbers greater than 2 do not have a phi(n) value of 12 because they are not relatively prime to any number.
No, phi(n) cannot be negative. By definition, phi(n) counts the number of positive integers less than or equal to n that are relatively prime to n, so it will always be a positive integer.
Phi(n) is related to prime numbers because prime numbers have a phi(n) value of (p-1), where p is the prime number. This is because all numbers less than p are relatively prime to p, and there are (p-1) of them.