## 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: $1512/480$