Your idea matters-1
What is the sum of the positive divisors of 5 0 4 0 ?
The answer is 19344.
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4 solutions
The question should specify that we are talking about positive divisors, or else the answer is 0.
The programming solution for this problem in Java:
public static void main(String[]args)
{
int num = 5040;
int sum = 0;
int factor = 1;
for (num = 5040;factor<=5040;)
{
if (num%factor==0)
{
sum+=factor;
}
factor++;
}
System.out.println(sum);
}
We can write,
5 0 4 0 = 2 4 × 3 2 × 5 × 7
So the sum of 5040's divisors is, ( 2 0 + 2 1 + 2 2 + 2 3 + 2 4 ) ( 3 0 + 3 1 + 3 2 ) ( 5 0 + 5 1 ) ( 7 0 + 7 1 ) = ( 1 + 2 + 4 + 8 + 1 6 ) ( 1 + 3 + 9 ) ( 1 + 5 ) ( 1 + 7 ) = 3 1 × 1 3 × 6 × 8 = 1 9 3 4 4
5 0 4 0 = 1 × 5 0 4 0 = 2 × 2 5 2 0 = 3 × 1 6 8 0 = 4 × 1 2 6 0 = 5 × 1 0 0 8 = 6 × 8 4 0 = 7 × 7 2 0 = 8 × 6 3 0 = 9 × 5 6 0 = 1 0 × 5 0 4 = 1 2 × 4 2 0 = 1 4 × 3 6 0 = 1 5 × 3 3 6 = 1 6 × 3 1 5 = 1 8 × 2 8 0 = 2 0 × 2 5 2 = 2 1 × 2 4 0 = 2 4 × 2 1 0 = 2 8 × 1 8 0 = 3 0 × 1 6 8 = 3 5 × 1 4 4 = 3 6 × 1 4 0 = 4 0 × 1 2 6 = 4 2 × 1 2 0 = 4 5 × 1 1 2 = 4 8 × 1 0 5 = 5 6 × 9 0 = 6 0 × 8 4 = 6 3 × 8 0 = 7 0 × 7 2
1 + 5 0 4 0 + 2 + 2 5 2 0 + 3 + 1 6 8 0 + 4 + 1 2 6 0 + 5 + 1 0 0 8 + 6 + 8 4 0 + 7 + 7 2 0 + 8 + 6 3 0 + 9 + 5 6 0 + 1 0 + 5 0 4 + 1 2 + 4 2 0 + 1 4 + 3 6 0 + 1 5 + 3 3 6 + 1 6 + 3 1 5 + 1 8 + 2 8 0 + 2 0 + 2 5 2 + 2 1 + 2 4 0 + 2 4 + 2 1 0 + 2 8 + 1 8 0 + 3 0 + 1 6 8 + 3 5 + 1 4 4 + 3 6 + 1 4 0 + 4 0 + 1 2 6 + 4 2 + 1 2 0 + 4 5 + 1 1 2 + 4 8 + 1 0 5 + 5 6 + 9 0 + 6 0 + 8 4 + 6 3 + 8 0 + 7 0 + 7 2 = 1 9 3 4 4
Since 5 0 4 0 = 2 4 × 3 2 × 5 × 7 the sum of its divisors is ( 1 + 2 + 4 + 8 + 1 6 ) ( 1 + 3 + 9 ) ( 1 + 5 ) ( 1 + 7 ) = 3 1 × 1 3 × 6 × 8 = 1 9 3 4 4 .