Are there 21 different positive whole numbers such that the sum of their reciprocals is 1?

:) I'm not sure about this question

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- Dec 7th 2007, 06:51 PMmr.torontoreciprocals
**Are there 21 different positive whole numbers such that the sum of their reciprocals is 1?**

:) I'm not sure about this question - Dec 7th 2007, 10:07 PMbadgerigarYes
Yes.

1/2+1/3 = 5/6

$\displaystyle 1/2+1/3+\frac{1/2+1/3}{6}$ = 35/36

$\displaystyle 1/2+1/3+\frac{1/2+1/3}{6}+\frac{\frac{1/2+1/3}{6}}{6}$ = $\displaystyle \frac{6^3-1}{6^3}$

We can keep doing this forever so the sum of any even number of reciprocals of natural numbers can represent $\displaystyle \frac{6^n-1}{6^n}$

Adding a last number of 1/6^n produces 1, so the sum of any odd number of reciprocals of 1atural numbers can be 1