Is 8128 a Perfect Number?

A perfect number is a positive integer that is equal to the sum of its proper divisors. Proper divisors of a number are the positive divisors excluding the number itself. In order to determine whether 8128 is a perfect number or not, we need to find its proper divisors and calculate their sum.

Calculating Proper Divisors of 8128:
To find the proper divisors of 8128, we need to find all the numbers that divide 8128 evenly without leaving a remainder. Let's calculate the proper divisors step by step:

1. Start by finding the prime factorization of 8128. Prime factorization is the process of breaking down a number into its prime factors.
- Prime factors of 8128: 2 * 2 * 2 * 2 * 127 = 2^4 * 127

2. Now, we can find all the divisors of 8128 by considering all the combinations of the prime factors.
- The divisors of 8128 are: 1, 2, 4, 8, 16, 32, 127, 254, 508, 1016, 2032, 4064

3. To find the proper divisors, we exclude the number itself, which means we remove 8128 from the list of divisors.
- The proper divisors of 8128 are: 1, 2, 4, 8, 16, 32, 127, 254, 508, 1016, 2032, 4064

Calculating the Sum of Proper Divisors:
To determine if 8128 is a perfect number, we need to sum up its proper divisors and check if the sum equals 8128.

- The sum of the proper divisors of 8128 is: 1 + 2 + 4 + 8 + 16 + 32 + 127 + 254 + 508 + 1016 + 2032 + 4064 = 8128

Conclusion:
Considering the definition of a perfect number, we have found that the sum of the proper divisors of 8128 is equal to the number itself. Therefore, 8128 is indeed a perfect number.

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