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Poirot1
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Show that g(n)=8n/15 iff n is divisible by 3 and 5 and by no other primes, where g is the euler totient function.
How to go about the proof?
Poirot said:
Show that g(n)=8n/15 iff n is divisible by 3 and 5 and by no other primes, where g is the euler totient function.
How to go about the proof?
The Totient function, denoted as φ(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.
The Totient function is calculated by finding all the positive integers less than or equal to n that are relatively prime to n. This can be done by finding the prime factorization of n and using the formula φ(n) = n * (1 - 1/p_{1}) * (1 - 1/p_{2}) * ... * (1 - 1/p_{k}), where p_{1}, p_{2}, ..., p_{k} are the distinct prime factors of n.
The Totient function has many applications in number theory, cryptography, and modular arithmetic. It is also used to solve various problems, such as finding the number of reduced fractions with a given denominator and determining the order of elements in a group.
The proof for the Totient function is based on Euler's theorem, which states that if a and n are coprime positive integers, then a raised to the power of φ(n) is congruent to 1 modulo n. This proof can be extended to all positive integers using the Chinese remainder theorem.
One common misconception is that the Totient function only works for prime numbers. However, it can be applied to any positive integer. Another misconception is that the Totient function is the same as the phi function, which is used in different branches of mathematics and has a slightly different definition.