Sums of Reciprocals

The sum of the positive divisors of 480 is 1512. Find the sum of the reciprocals of the positive divisors of 480.
Source: NCTM Mathematics Teacher, February 2008

Solution
Given that 480=2^5\cdot 3^1\cdot 5^1, the number of divisors is (5+1)(1+1)(1+1)=24. The first twelve divisors are small and easy to guess: 1,2,3,4,5,6,8,10,12,15,16, and 20 and as a bonus we get the last twelve by dividing 480 by the first twelve divisors. For example, 480/1=480,480/2=240,480/3=160, etc. : 480,240,160,120,96,80,60,48,40,32,30,24.
When we add the reciprocals of the divisors, the divisors appear as denominators in a set of 24 fractions
\dfrac{1}{1}+\dfrac{1}{2}+\dfrac{1}{3}+\cdots+\dfrac{1}{240}+\dfrac{1}{480}
When we reduce the fractions to 480 (the least common denominator), the 24 divisors appear as numerators
\dfrac{480}{480}+\dfrac{240}{480}+\cdots+\dfrac{2}{480}+\dfrac{1}{480}=\dfrac{480+240+\cdots+2+1}{480}=\dfrac{1512}{480}

Answer: 1512/480

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About mvtrinh

Retired high school math teacher.
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