Proving Euler's Phi Function: Any Help Appreciated!

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SUMMARY

The discussion centers on proving the relationship between Euler's phi function φ(m) and the count of integers n such that 1≤n≤mk and (n,m)=1, specifically that this count equals kφ(m). Participants emphasize the modular nature of coprime integers, noting that if c is coprime to m, then c + km remains coprime to m. This modular repetition of coprimes is crucial for understanding the proof of the stated relationship.

PREREQUISITES
  • Understanding of Euler's phi function φ(m)
  • Familiarity with coprime integers and their properties
  • Basic knowledge of modular arithmetic
  • Experience with the division algorithm
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  • Study the properties of Euler's phi function in detail
  • Learn about the division algorithm and its applications in number theory
  • Explore modular arithmetic and its implications for coprime integers
  • Investigate proofs related to the distribution of coprime integers
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Mathematicians, students of number theory, and anyone interested in understanding the properties of coprime integers and Euler's phi function.

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"Let m and k be positive integers and φ(m) is the Euler's phi function. Then the number of integers n such that 1≤n≤mk and (n,m)=1 is kφ(m)."


I can't figure out why this is true. How can we prove it?

Can someone explain this, please?
Any help is appreciated!
 
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Coprimes repeat in a kind of "modular" fashion. You may try using the division algorithm on n, and expressing it as n=qm+r; then try to prove that (n,m)=(r,m).
 
hmm...sorry I don't understand. What do you mean by "Coprimes repeat in a kind of 'modular' fashion"?
 
kingwinner said:
hmm...sorry I don't understand. What do you mean by "Coprimes repeat in a kind of 'modular' fashion"?

If c is coprime to m, then c+m is coprime to m, and in general c+km is coprime to m.
 

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