Find All Integers Such that phi(n)=12

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In summary, the conversation discusses finding all integers where phi(n)=12, specifically mentioning 13 as one possible solution and inquiring about composite numbers. The conversation also touches on the relationship between phi(a) and phi(b) in regard to phi(ab), as well as the formula for phi(p^n) if p is prime. The question of finding all combinations of n where phi(n) add to 12 is also raised.
  • #1
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?
-Thanks
 
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  • #2
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?
 
  • #3
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?

[tex]\varphi(a)\varphi(b)[/tex]=[tex]\varphi(ab)[/tex] if (a,b)=1. [tex]\varphi(p^n)[/tex]= [tex]\(p^n)[/tex]-[tex]\(p^n-1)[/tex] if p is prime. So am I looking for all combinations of n in which the respective phi(n) add to equal 12? I.E. am I searching for prime factorizations of some n where these two properties will yield of phi of 12?
 

1. What is the definition of phi(n)?

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.

2. How do you find all integers such that phi(n) = 12?

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.

3. Are there any integers that do not have a phi(n) value of 12?

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.

4. Can phi(n) ever be negative?

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.

5. How is phi(n) related to prime numbers?

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.

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