Question 13

Number Theory Level 3

We call a proper divisor of an integer N N is a positive divisor of N N that is less than N N . For example, the proper divisors of 12 are 1, 2, 3, 4 and 6.

What is the sum of the proper divisors of the sum of the proper divisors of 284?


The answer is 284.

Trevor B. by Trevor B. , 22, USA

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3 solutions

Reeshabh Ranjan
Jun 18, 2014

Proper divisors of 284 are 1, 2, 4, 71 and 142. They add up to 220. The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110. They add up to 284.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Because 220 and 284 are amicable numbers

tasmeem reza - 6 years, 10 months ago
Bill Bell
Jul 30, 2014

Using the Python sympy library:

sum(divisors(sum(divisors(284)[:-1]))[:-1])

0 Helpful 0 Interesting 0 Brilliant 0 Confused

nonsense question you have asked only the sum of proper divisors of 284

S rohith - 6 years, 9 months ago
Vishal Barman
Jul 30, 2014

First,find the proper divisors of 284: 

  1, 2, 4, 71, 142

These sum to 220. Then, find the proper divisors of 220: 

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

If you add all these numbers together, you can see they sum to 284. Thus, 220 and 284 are a pair of amicable numbers.

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