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4th Edition

Richard N. Aufmann + 3 others

Publisher: Cengage Learning

ISBN: 9781305965584

Chapter 6.6, Problem 39ES

Textbook Problem

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**A Sum of Reciprocals Theorem** The sum of the reciprocals of all the positive divisors of a perfect number is always 2.

Verify this theorem for each of the following perfect numbers.

- 6
- 28

To determine

**(a)**

**To Verify:**

The sum of reciprocals theorem for the perfect number

**Given information:**

The sum of reciprocals theorem states that the sum of the reciprocals of all the positive divisors of a perfect number is always 2.

**Formula used:**

Find the sum of reciprocals of all the positive divisors of the given perfect number.

**Proof:**

Consider the perfect number

To determine

**(b)**

**To Verify:**

The sum of reciprocals theorem for the perfect number

Mathematical Excursions (MindTap Course List)

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Ch. 6.5 -
Ch. 6.5 -
Ch. 6.5 - The RSA Algorithm In 1977, Ron Rivest, Adi Shamir,...Ch. 6.5 -
Ch. 6.6 - Use the above technique to find the Sum of the...Ch. 6.6 - Use the above technique to find the Sum of the...Ch. 6.6 - Use the above technique to find the Sum of the...Ch. 6.6 - Use the above technique to find the Sum of the...Ch. 6.6 - Use deductive reasoning to prove that every prime...Ch. 6.6 - Use inductive reasoning to decide whether every...Ch. 6.6 - Determine whether each number is perfect,...Ch. 6.6 - Determine whether each number is perfect,...Ch. 6.6 - Determine whether each number is perfect,...Ch. 6.6 - Determine whether each number is perfect,...Ch. 6.6 - Determine whether each number is perfect,...Ch. 6.6 - Determine whether each number is perfect,...Ch. 6.6 - Determine whether each number is perfect,...Ch. 6.6 - Determine whether each number is perfect,...Ch. 6.6 - Determine whether each number is perfect,...Ch. 6.6 - Determine whether each number is perfect,...Ch. 6.6 - Determine whether each number is perfect,...Ch. 6.6 - Determine whether each number is perfect,...Ch. 6.6 - Determine whether each number is perfect,...Ch. 6.6 - Determine whether each number is perfect,...Ch. 6.6 - Determine whether each number is perfect,...Ch. 6.6 - Determine whether each number is perfect,...Ch. 6.6 - Determine whether each Mersenne number is a prime...Ch. 6.6 - Determine whether each Mersenne number is a prime...Ch. 6.6 - Determine whether each Mersenne number is a prime...Ch. 6.6 - Determine whether each Mersenne number is a prime...Ch. 6.6 - In 1876, Ä‚â€°douard Lucas proved, without the aid of...Ch. 6.6 - In 1952, Raphael M. Robinson proved, with the aid...Ch. 6.6 - Determine the number of digits in the given...Ch. 6.6 - Determine the number of digits in the given...Ch. 6.6 - Determine the number of digits in the given...Ch. 6.6 - Determine the number of digits in the given...Ch. 6.6 - Determine the number of digits in the given...Ch. 6.6 - Determine the number of digits in the given...Ch. 6.6 - Determine the number of digits in the given...Ch. 6.6 - Determine the number of digits in the given...Ch. 6.6 - Verify that x=9,y=15, and n=5 do not yield a...Ch. 6.6 - Verify that x=7,y=19, and n=6 do not yield a...Ch. 6.6 - Determine whether each of the following statements...Ch. 6.6 - Show that 4078n+3433n=12,406n cannot be a solution...Ch. 6.6 - Fermat's Little Theorem A theorem known as...Ch. 6.6 - Amicable Numbers The Greeks considered the pair of...Ch. 6.6 - A Sum of Cubes Property The perfect number 28 can...Ch. 6.6 - A Sum of the Digits Theorem If you sum the digits...Ch. 6.6 - A Sum of Reciprocals Theorem The sum of the...Ch. 6.6 - The Smallest Odd Abundant Number Determine the...Ch. 6.6 - Fermat Numbers Numbers of the form 22n+1, where n...Ch. 6.6 - Semiperfect Numbers Any number that is the sum of...Ch. 6.6 - Weird Numbers Any number that is an abundant...Ch. 6.6 - A False Prediction In 1811, Peter Barlow wrote in...Ch. 6 - Write 4,506,325 using Egyptian hieroglyphics.Ch. 6 - Write 3,124,043 using Egyptian hieroglyphics.Ch. 6 - Write the Egyptian hieroglyphic as a Hindu-Arabic...Ch. 6 - Write the Egyptian hieroglyphic as a Hindu-Arabic...Ch. 6 - Write each Roman numeral as a Hindu-Arabic...Ch. 6 - Write each Roman numeral as a Hindu-Arabic...Ch. 6 - Write each Roman numeral as a Hindu-Arabic...Ch. 6 - Write each Roman numeral as a Hindu-Arabic...Ch. 6 - Write each Hindu-Arabic numeral as a Roman...Ch. 6 - Write each Hindu-Arabic numeral as a Roman...Ch. 6 - Write each Hindu-Arabic numeral as a Roman...Ch. 6 - Write each Hindu-Arabic numeral as a Roman...Ch. 6 - Write each Hindu-Arabic numeral in expanded form....Ch. 6 - Write each Hindu-Arabic numeral in expanded form....Ch. 6 - Simplify each expanded form....Ch. 6 - Simplify each expanded form....Ch. 6 - Write each Babylonian numeral as a Hindu-Arabic...Ch. 6 - Write each Babylonian numeral as a Hindu-Arabic...Ch. 6 - Write each Babylonian numeral as a Hindu-Arabic...Ch. 6 - Write each Babylonian numeral as a Hindu-Arabic...Ch. 6 - Write each Hindu-Arabic numeral as Baylonian...Ch. 6 - Write each Hindu-Arabic numeral as Baylonian...Ch. 6 - Write each Hindu-Arabic numeral as Baylonian...Ch. 6 - Write each Hindu-Arabic numeral as Baylonian...Ch. 6 - Write each Mayan numeral as a Hindu-Arabic...Ch. 6 - Write each Mayan numeral as a Hindu-Arabic...Ch. 6 - Write each Mayan numeral as a Hindu-Arabic...Ch. 6 - Write each Mayan numeral as a Hindu-Arabic...Ch. 6 - Write each Hindu-Arabic numeral as Mayan numeral....Ch. 6 - Write each Hindu-Arabic numeral as Mayan numeral....Ch. 6 - Write each Hindu-Arabic numeral as Mayan numeral....Ch. 6 - Write each Hindu-Arabic numeral as Mayan numeral....Ch. 6 - Convert each numeral to base ten. 45sixCh. 6 - Convert each numeral to base ten. 172nineCh. 6 - Convert each numeral to base ten. E3sixteenCh. 6 - Convert each numeral to base ten. 1BAtwelveCh. 6 - Convert each base ten numeral to the indicated...Ch. 6 - Convert each base ten numeral to the indicated...Ch. 6 - Convert each base ten numeral to the indicated...Ch. 6 - Convert each base ten numeral to the indicated...Ch. 6 - Convert each numeral to the indicated base....Ch. 6 - Convert each numeral to the indicated base....Ch. 6 - Convert each numeral to the indicated base....Ch. 6 - Convert each numeral to the indicated base....Ch. 6 - Convert each numeral directly (without first...Ch. 6 - Convert each numeral directly (without first...Ch. 6 - Convert each numeral directly (without first...Ch. 6 - Convert each numeral directly (without first...Ch. 6 - Convert each numeral directly (without first...Ch. 6 - Convert each numeral directly (without first...Ch. 6 - Convert each numeral directly (without first...Ch. 6 - Convert each numeral directly (without first...Ch. 6 - Use the double-dabble method to convert each base...Ch. 6 - Use the double-dabble method to convert each base...Ch. 6 - Use the double-dabble method to convert each base...Ch. 6 - Use the double-dabble method to convert each base...Ch. 6 - Perform the indicated operation. Write the answers...Ch. 6 - Perform the indicated operation. Write the answers...Ch. 6 - Perform the indicated operation. Write the answers...Ch. 6 - Perform the indicated operation. Write the answers...Ch. 6 - Perform the indicated operation. Write the answers...Ch. 6 - Perform the indicated operation. Write the answers...Ch. 6 - Perform the indicated operation. Write the answers...Ch. 6 - Perform the indicated operation. Write the answers...Ch. 6 - Use divisibility tests to determine whether the...Ch. 6 - Use divisibility tests to determine whether the...Ch. 6 - Determine whether the given number is a prime...Ch. 6 - Determine whether the given number is a prime...Ch. 6 - Determine whether the given number is a prime...Ch. 6 - Determine whether the given number is a prime...Ch. 6 - Determine the prime factorization of the given...Ch. 6 - Determine the prime factorization of the given...Ch. 6 - Determine the prime factorization of the given...Ch. 6 - Determine the prime factorization of the given...Ch. 6 - Determine whether the given number is perfect,...Ch. 6 - Determine whether the given number is perfect,...Ch. 6 - Determine whether the given number is perfect,...Ch. 6 - Determine whether the given number is perfect,...Ch. 6 - Use Euclid's perfect number-generating procedure...Ch. 6 - Use Euclid's perfect number-generating procedure...Ch. 6 - Use the Egyptian doubling procedure to find the...Ch. 6 - Use the Egyptian doubling procedure to find the...Ch. 6 - Use the Egyptian doubling procedure to find the...Ch. 6 - Use the Egyptian doubling procedure to find the...Ch. 6 - Find the number of digits in the Mersenne number...Ch. 6 - Find the number of digits in the Mersenne number...Ch. 6 - How many Odd perfect numbers had been discovered...Ch. 6 - Determine whether there is a natural number z such...Ch. 6 - Write 3124 using Egyptian hieroglyphics.Ch. 6 - Write the Egyptian hieroglyphic as a Hindu-Arabic...Ch. 6 - Write the Roman numeral MCDXLVII as a Hindu-Arabic...Ch. 6 - Write 2609 as Roman numeral.Ch. 6 - Write 67,485 in expanded form.Ch. 6 - Simplify: (5105)+(3104)+(2102)+(8101)+(4100)Ch. 6 - Write the Babylonian numeral as a Hindu-Arabic...Ch. 6 - Write 9675 as a Babylonian numeral.Ch. 6 - Write the Mayan numeral as a Hindu-Arabic numeral.Ch. 6 - Write 502 as a Mayan numeral.Ch. 6 - Convert 3542six to base ten.Ch. 6 - Convert 2148 to a. base eight and b. base twelve.Ch. 6 - Convert 4567eight to binary form.Ch. 6 - Convert 101010110111two to hexadecimal form.Ch. 6 - Perform the indicated operation. Write the answer...Ch. 6 - Perform the indicated operation. Write the answer...Ch. 6 - Perform the indicated operation. Write the answer...Ch. 6 - Perform the indicated operation. Write the answer...Ch. 6 - Determine the prime factorization of 230.Ch. 6 - Determine whether 1001 is a prime number or a...Ch. 6 - Use divisibility tests to determine whether...Ch. 6 - Use divisibility test to determine whether...Ch. 6 - Determine whether 96 is perfect, deficient, or...Ch. 6 - Use Euclid's perfect number generating procedure...Ch. 6 - Explain how you know (without doing any...

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