Proving if it is possible to write 1 as the sum of the reciprocals of x odd integers
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.
Solution 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.