Prove that the Diophantine equation has only one solution for $a,b,c$. Prove that the Diophantine equation
$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$     
has only one solution for positive integers $a,b,c$   with $a<b<c$

Hint:
$$1\geq\frac{1}{a}>\frac{1}{b}>\frac{1}{c}$$ so , $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}<\frac{3}{a}$$.
If $a>3$ the sum is less than $1$.
Check the cases $a=2$ , $a=1$.
