Prove it should have no solution

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SUMMARY

The discussion centers on the mathematical proof that the equation $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{d}+\dfrac{1}{e}+\dfrac{1}{f}=1$ has no solution when $a, b, c, d, e, f$ are all odd natural numbers. Participants provided rigorous logical arguments and explored the implications of the properties of odd numbers in relation to their reciprocals. The consensus is that the sum of the reciprocals of six odd natural numbers cannot equal one, confirming the assertion definitively.

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Albert1
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$a,b,c,d,e,f\in N $and all of them are odd numbers

prove :

$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{d}+\dfrac{1}{e}+\dfrac{1}{f}=1$ has no solution
 
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Suppose there are odd $a, b, c, d, e, f$ satisfying the equation. Then they also satisfy:
$$bcdef + acdef + abdef + abcef + abcdf + abcde = abcdef$$
The LHS is a sum of six odd terms (products of odd numbers) so it is even, but the RHS is odd, which is a contradiction.
$\blacksquare$
 

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