### Instruction: What is the sum of the proper divisors of 256?### Response: A proper divisor of a number is a positive factor other than the number itself. For example, the proper divisors of 12 are 1, 2, 3, 4, and 6.
To find the proper divisors of 256, I can use prime factorization.
256 is a power of 2, so I can write it as 2^8.
This means that any proper divisor of 256 must be of the form 2^k, where k is an integer between 0 and 7.
The proper divisors of 256 are 2^0, 2^1, 2^2, ..., 2^7, which are 1, 2, 4, ..., 128.
To find the sum of these divisors, I can use the formula for the sum of a geometric series.
The sum of a geometric series with first term a, common ratio r, and n terms is a(1 - r^n) / (1 - r).