Sum of Divisors

Little Rajesh, a 7th standard Math genius has a question for you. He knows for sure that you cannot answer this question but at least try. The question is this: Find the sum of all divisors of 43,518.


Hint 1

The problem can be solved with no formula within 5 minutes.


Hint 2

Try to find the sum of all divisors of 5. Apply the same technique to this problem also.


Hint 3

The answer is same for any integer.


Answer

Zero.


Solution

In Mathematics, the definition of a divisor is A Divisor of an integer N, also called a factor of N, is an integer which divides N without leaving a remainder. The definition implies that negative divisors are also there.

Consider any integer, say, 12. The positive divisors of 12 are 1, 2, 3, 4, 6 and 12. The negative divisors of 12 are -1, -2, -3, -4, -6 and -12. So, when we add all the divisors of 12, we get zero. By the same reasoning, the sum of all divisors of 43,518 is zero.

Generalizing the problem, if n is any integer, the sum of all divisors of n is zero.