Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 5 - Number Theory and the Real Number System - 5.1 Number Theory: Prime and Composite Numbers - Exercise Set 5.1 - Page 258: 124

Answer

Perfect numbers are defined as the sum of the proper divisors except number should be equal to that positive number and perfect numbers can be determined using Euclid theorem then the perfect number will be equal to \[{{2}^{n-1}}\left( {{2}^{n}}-1 \right)\].

Work Step by Step

Perfect number: Perfect numbers are defined as the sum of the proper divisors except number should be equal to that positive number and perfect numbers can be determined using Euclid theorem, then the perfect number will be equal to \[{{2}^{n-1}}\left( {{2}^{n}}-1 \right)\]. Here, \[n\]is the prime number and \[n>1\]. Now, substitute the value of n to find perfect numbers. Then, If \[n=3\] then, a perfect number is, \[\begin{align} & {{2}^{2-1}}\left( {{2}^{2}}-1 \right)=2\times 3 \\ & =6 \end{align}\] So, the perfect number is \[6\], And, the divisors of this number are, \[6=1\times 2\times 3\times 6\] Now, find the sum of the divisors except the number. The sum of divisors is \[\left( 1+2+3=6 \right)\]. If \[n=5\] then, a perfect number is, \[\begin{align} & {{2}^{5-1}}\left( {{2}^{5}}-1 \right)=16\times 31 \\ & =496 \end{align}\] So, the perfect number is \[496\]. And, the divisors of this number are: \[496=1,2,4,8,16,31,62,124,248,496\] Now, find the sum of the divisors except the number. The sum of divisors is \[\left( 1+2+4+8+16+31+62+124+248=496 \right)\]. Hence, the above examples are perfect numbers.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.