5. Show that o (n) is odd if n is a power of two. 6. Prove that if f(n) is multiplicative, then so is f(n)/n. 7 What is the smallest integer n such that d(n) = 8? Such that d(n) = 10
5. Show that o (n) is odd if n is a power of two. 6. Prove that if f(n) is multiplicative, then so is f(n)/n. 7 What is the smallest integer n such that d(n) = 8? Such that d(n) = 10
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
6&13
![## Problems
1. Calculate \( d(42), \sigma(42), d(420), \) and \( \sigma(420) \).
2. Calculate \( d(540), \sigma(540), d(5400), \) and \( \sigma(5400) \).
3. Calculate \( d \) and \( \sigma \) of 10115, given:
\[
10115 = 5 \cdot 7 \cdot 17^2 \quad \text{and} \quad 100115 = 5 \cdot 20023.
\]
4. Calculate \( d \) and \( \sigma \) of 10116, given:
\[
10116 = 2^2 \cdot 3^2 \cdot 281 \quad \text{and} \quad 100116 = 2^2 \cdot 3^5 \cdot 103.
\]
5. Show that \( \sigma(n) \) is odd if \( n \) is a power of two.
6. Prove that if \( f(n) \) is multiplicative, then so is \( \frac{f(n)}{n} \).
7. What is the smallest integer \( n \) such that \( d(n) = 8 \)? Such that \( d(n) = 10 \)?
8. Does \( d(n) = k \) have a solution \( n \) for each \( k \)?
9. In 1644, Mersenne asked for a number with 60 divisors. Find one smaller than 10,000.
10. Find infinitely many \( n \) such that \( d(n) = 60 \).
11. If \( p \) is an odd prime, for which \( k \) is \( 1 + p + \cdots + p^k \) odd?
12. For which \( n \) is \( \sigma(n) \) odd?
13. If \( n \) is a square, show that \( d(n) \) is odd.
14. If \( d(n) \) is odd, show that \( n \) is a square.
15. Observe that:
\[
1 + \frac{1}{3} = \frac{4}{3}; \quad 1 + \frac{1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbe7e575d-dfd7-411c-ad5f-62e2be64ba9f%2Fd2455b01-c91a-488e-8e3b-47ea365c4f96%2Fidaujiv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Problems
1. Calculate \( d(42), \sigma(42), d(420), \) and \( \sigma(420) \).
2. Calculate \( d(540), \sigma(540), d(5400), \) and \( \sigma(5400) \).
3. Calculate \( d \) and \( \sigma \) of 10115, given:
\[
10115 = 5 \cdot 7 \cdot 17^2 \quad \text{and} \quad 100115 = 5 \cdot 20023.
\]
4. Calculate \( d \) and \( \sigma \) of 10116, given:
\[
10116 = 2^2 \cdot 3^2 \cdot 281 \quad \text{and} \quad 100116 = 2^2 \cdot 3^5 \cdot 103.
\]
5. Show that \( \sigma(n) \) is odd if \( n \) is a power of two.
6. Prove that if \( f(n) \) is multiplicative, then so is \( \frac{f(n)}{n} \).
7. What is the smallest integer \( n \) such that \( d(n) = 8 \)? Such that \( d(n) = 10 \)?
8. Does \( d(n) = k \) have a solution \( n \) for each \( k \)?
9. In 1644, Mersenne asked for a number with 60 divisors. Find one smaller than 10,000.
10. Find infinitely many \( n \) such that \( d(n) = 60 \).
11. If \( p \) is an odd prime, for which \( k \) is \( 1 + p + \cdots + p^k \) odd?
12. For which \( n \) is \( \sigma(n) \) odd?
13. If \( n \) is a square, show that \( d(n) \) is odd.
14. If \( d(n) \) is odd, show that \( n \) is a square.
15. Observe that:
\[
1 + \frac{1}{3} = \frac{4}{3}; \quad 1 + \frac{1
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