Question

The proper divisors of 12 are 1, 2, 3, 4 and 6. A proper divisor of an integer $N$ is a positive divisor of $N$ that is less than $N$. What is the sum of the proper divisors of the sum of the proper divisors of 284

Answer 1

284

Explanation:

The proper divisors of 284 are ...

1, 2, 4, 71, 142

Their sum is 220.

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The proper divisors of 220 are ...

1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110

Their sum is 284.

The sum of the proper divisors of the sum of the proper divisors of 284 is 284.

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Additional comment

For the prime factorization of a number N, the number of divisors (proper and improper) of N is the product of the exponents of the prime factors, each exponent being increased by 1.

Example:

The prime factorization of 220 is 2²·5·11, so the exponents of the prime factors are 2, 1, 1. The number of divisors of 220 is (2+1)(1+1)(1+1) = 12. 11 of those are the proper divisors listed above. The 12th one is 220.