\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
A perfect number is a natural integer that is equal to the sum of its proper divisors (all its positive divisors except itself).
Consider the number \(28\). Its positive divisors are \(1, 2, 4, 7, 14, 28\).
Its proper divisors are \(1, 2, 4, 7, 14\).
Since \(1 + 2 + 4 + 7 + 14 = 28\), the number 28 is a perfect number.
Euclid provided the following rule to find perfect numbers:
"If \(a = 2^n(2^{n+1} - 1)\) and if \((2^{n+1} - 1)\) is prime, then \(a\) is perfect."
  1. Find the first three perfect numbers using Euclid's rule.
  2. Let \(a = 2^n(2^{n+1} - 1)\) where \(q = 2^{n+1} - 1\) is a prime number.
    1. What is the prime factorization of \(a\)?
    2. Deduce the complete list of divisors of \(a\).
    3. Prove that the sum of the proper divisors of \(a\) is equal to \(a\).

Capture an image of your work. AI teacher feedback takes approximately 10 seconds.