What is the smallest positive integer whose proper divisors add up to more than the integer itself?

The smallest positive integer whose proper divisors add up to more than the integer itself is 12.

Here's the step-by-step explanation:

1. A proper divisor is a positive integer that divides another integer evenly, excluding the integer itself. For example, the proper divisors of 10 are 1, 2, and 5.

2. We need to find the smallest positive integer whose sum of proper divisors is greater than the integer itself. This means we are looking for a number that is the smallest "abundant number". An abundant number is a number that the sum of its proper divisors is greater than the number itself.

3. Let's start checking from the smallest positive integer, which is 1. The proper divisor of 1 is none (excluding 1 itself), so the sum is 0.

4. The proper divisors of 2 are 1, the sum is 1.

5. The proper divisors of 3 are 1, the sum is 1.

6. The proper divisors of 4 are 1 and 2, the sum is 3.

7. The proper divisors of 5 are 1, the sum is 1.

8. The proper divisors of 6 are 1, 2, and 3, the sum is 6.

9. The proper divisors of 7 are 1, the sum is 1.

10. The proper divisors of 8 are 1, 2, and 4, the sum is 7.

11. The proper divisors of 9 are 1 and 3, the sum is 4.

12. The proper divisors of 10 are 1, 2, and 5, the sum is 8.

13. The proper divisors of 11 are 1, the sum is 1.

14. The proper divisors of 12 are 1, 2, 3, 4, and 6, the sum is 16.

So, the smallest positive integer whose proper divisors add up to more than the integer itself is 12.