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13. If n is a square, show that d (n) is odd. 14 TC

13. If n is a square, show that d (n) is odd. 14 TC

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Problems * 1. Calculate d(42), o(42), d(420), and o(420). 2. Calculate d (540), o(540), d(5400), and o(5400). * 3. Calculated and o of 10115 = 5 · 7 · 17² and 100115 = 5. 20023. 4. Calculated and o of 10116 = 2² · 3² · 281 and 100116 = 2².35.103. 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? 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) 11. If p is an odd prime, for which k is 1 + p + · 12. For which n is o(n) odd? = 60. + pk 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 + 1/3 = 4/3; 1 + 1/2 + 1/4 = 7/4; 1 + 1/5 = 6/5; 1 + 1/2 +
Transcribed Image Text:Problems * 1. Calculate d(42), o(42), d(420), and o(420). 2. Calculate d (540), o(540), d(5400), and o(5400). * 3. Calculated and o of 10115 = 5 · 7 · 17² and 100115 = 5. 20023. 4. Calculated and o of 10116 = 2² · 3² · 281 and 100116 = 2².35.103. 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? 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) 11. If p is an odd prime, for which k is 1 + p + · 12. For which n is o(n) odd? = 60. + pk 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 + 1/3 = 4/3; 1 + 1/2 + 1/4 = 7/4; 1 + 1/5 = 6/5; 1 + 1/2 +
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