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saadsarfraz
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Q- Let m and n be coprime. Show that\phi(mn) = \phi(m) * \phi(n). Hint: when does a pair of residues modulo m and n have an inverse.
What is the Euler Totient Function for Coprime Numbers?
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SUMMARY
The Euler Totient Function for coprime numbers states that if m and n are coprime, then the relationship \(\phi(mn) = \phi(m) \times \phi(n)\) holds true. This is derived from the structure of the multiplicative group of integers modulo mn, represented as \((\mathbf{Z}/mn\mathbf{Z})^{\times} = (\mathbf{Z}/m\mathbf{Z})^{\times} \times (\mathbf{Z}/n\mathbf{Z})^{\times}\). The proof involves analyzing the orders of both sides of the equation, confirming that the totient function behaves multiplicatively for coprime integers.
PREREQUISITES- Understanding of the Euler Totient Function (\(\phi\))
- Knowledge of coprime integers and their properties
- Familiarity with modular arithmetic
- Basic concepts of group theory in number theory
- Study the properties of the Euler Totient Function in detail
- Explore modular arithmetic and its applications in number theory
- Learn about the structure of multiplicative groups in number theory
- Investigate other number-theoretic functions and their relationships
Mathematicians, students of number theory, and anyone interested in the properties of coprime integers and the Euler Totient Function.
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