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Combinatorial Study of φ(n), d(n) and σ(n).

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Arithmetic Functions
Combinatorial Study of &#966;(n)

1. If gcd(m,n) = 1, then &#966;(m,n) = &#966;(m)&#966;(n).
Use this to give a proof that &#966;(n) = n &#928;(1 - 1/p)
p/n

2. Prove that d(n) is odd iff n is a perfect square.

3. Prove that &#963;(n) &#8801; d(m)(mod 2) where m is the largest odd factor of n.

3.(2nd Part)
If &#963;(n) = 2n, n is a perfect number. Prove that if n is a perfect number , then
&#8721;1/d = 2.
d/n

4. Evaluate &#963;(210), &#966;(100) and &#963;(999).

5. Evaluate d(47), d(63) and d(150).

See the attached file.

Solution Summary

This solution is comprised of a detailed explanation of the Combinatorial Study of &#966;(n), d(n) and &#963;(n).
It contains step-by-step explanation for the following problem:

1. If gcd(m,n) = 1, then &#966;(m,n) = &#966;(m)&#966;(n).
Use this to give a proof that &#966;(n) = n &#928;(1 - 1/p)
p/n

2. Prove that d(n) is odd iff n is a perfect square.

3.Prove that &#963;(n) &#8801; d(m)(mod 2) where m is the largest odd factor of n.

3.(2nd Part)
If &#963;(n) = 2n, n is a perfect number. Prove that if n is a perfect number , then
&#8721;1/d = 2.
d/n

4. Evaluate &#963;(210), &#966;(100) and &#963;(999).

5. Evaluate d(47), d(63) and d(150).

Solution contains detailed step-by-step explanation.

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Arithmetic Functions
Combinatorial Study of &#966;(n)

1. If gcd(m,n) = 1, then &#966;(m,n) = &#966;(m)&#966;(n).
Use this ...

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• BSc, Manipur University
• MSc, Kanpur University
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