Find All Integers Such that phi(n)=12

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This discussion focuses on identifying all integers n such that the Euler's totient function φ(n) equals 12. The user confirms that n=13 is a valid solution and seeks methods to find composite numbers that satisfy this condition. Key properties discussed include the multiplicative nature of φ for coprime integers, expressed as φ(a)φ(b) = φ(ab), and the formula for prime powers, φ(p^n) = p^n - p^(n-1). The inquiry emphasizes the need to explore combinations of prime factorizations that yield φ(n) = 12.

PREREQUISITES
  • Understanding of Euler's totient function φ(n)
  • Knowledge of prime factorization
  • Familiarity with properties of coprime integers
  • Basic concepts of number theory
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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?
-Thanks
 
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What can you say about \varphi(a)\varphi(b) with regard to \varphi(ab)? What about \varphi(p^n) if p is prime?
 
Tinyboss said:
What can you say about \varphi(a)\varphi(b) with regard to \varphi(ab)? What about \varphi(p^n) if p is prime?

\varphi(a)\varphi(b)=\varphi(ab) if (a,b)=1. \varphi(p^n)= \(p^n)-\(p^n-1) 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?
 

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