Exercise · ZW-PERFECT-NUMBER-010 · ★★★ Advanced
Let \(n=2^{10}(2^{11}-1)\). Is \(n\) perfect?
Choose one option.
Show answer
- C. No, it is not perfect, because \(2^{11}-1=2047=23\cdot 89\) is not prime.
Explanation
Euclid's form requires \(2^p-1\) to be prime; here \(2047=23\cdot 89\), so the form does not apply and \(n\) is not perfect. Being even alone is not enough.
Review the concept: Perfect number
A perfect number is a positive integer \(n\) equal to the sum of its proper divisors, those smaller than \(n\); with the divisor-sum function \(\sigma\), the sum of all its divisors is twice \(n\).
\[\sigma(n)=2n\]
More practice problems: perfect number
- ZW-PERFECT-NUMBER-001 Which of these numbers is perfect?
- ZW-PERFECT-NUMBER-002 What is the second smallest perfect number?
- ZW-PERFECT-NUMBER-003 Which of these values of n satisfies the definition of a perfect number?
- ZW-PERFECT-NUMBER-004 How much do the proper divisors of 28 sum to?
- ZW-PERFECT-NUMBER-005 Which of these statements defines a perfect number n?
- ZW-PERFECT-NUMBER-006 Compute the sum of the proper divisors of 12 and state whether 12 is perfect.
- ZW-PERFECT-NUMBER-007 What is the third smallest perfect number?
- ZW-PERFECT-NUMBER-008 If 2^p-1 is prime with p=7, which perfect number does Euclid's formula give?