What is the sum of the reciprocals of the divisors of 360?

Answer 1

The answer is 3.25 or 13/4 First, notice that if you multiply the reciprocal of a divisor by 360, it's the same as dividing 360 by the divisor, so you get another divisor of 360. This means that if we multiply the sum we want by 360, each reciprocal-of-a-divisor becomes a divisor, and we get the sum of divisors of 360. This is far easier to calculate, so we'll find the sum of the divisors first, and then divide by 360. The prime factorization of 360 is 2*2*2*3*3*5, so the divisors are: 1, 2, 2*2, 2*2*2,

3, 2*3, 2*2*3, 2*2*2*3,

3*3, 2*3*3, 2*2*3*3, 2*2*2*3*3,

5, 2*5, 2*2*5, 2*2*2*5,

3*5, 2*3*5, 2*2*3*5, 2*2*2*3*5,

3*3*5, 2*3*3*5, 2*2*3*3*5, 2*2*2*3*3*5.

The way I have grouped them is for a reason. First, the second group is 5 times the first group, so the sum of the divisors is (1+5) times the first group. Second, in the first group, the second row is 3 times the first, and the third row is 9 times the first, so the sum of the first group is (1+3+9) times the first row. Finally, the sum of the first row is (1+2+4+8). So the sum of all the divisors is (1+5)(1+3+9)(1+2+4+8) = 6*13*15 = 1170. Now to get our final answer we have to divide by 360, getting 3.25. Of course, we could also get the answer by listing all the reciprocals and adding them. But this method works exactly the same way for any number, provided you can factor it. ----------------------------------------- New Question ---------------------------------------------- How would we get the fraction 360/1170 in this problem?

Answer 2

71/30