MHB Prove it should have no solution

[Albert1]

$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

 

[Nono713]

Spoiler

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.
$\blacksquare$