Let $x$ be an even number. Is it possible to write 1 as the sum of the reciprocals of $x$ odd integers? Write a proof supporting your answer.

I tried a lot of these, and I think it is no because I didn't find any possible combinations.


1 Answer 1


You can use contradiction to prove this. Suppose $$\frac1{k_1}+\frac1{k_2}+...\frac1{k_x}=1$$

Multiplying both sides by the denominators, you get $$k_2k_3...k_x+k_1k_3...k_x+...k_1k_2...k_{x-1}=k_1k_2...k_x$$

The left side is even but the right side is odd, and there's your contradiction.

  • 1
    $\begingroup$ @ suomynonA - (1+) (: foorp eciN $\endgroup$ Commented Oct 27, 2016 at 18:10

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