Consider the equation

\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1

where

a

b

and

c

are natural numbers and

0 < a < b < c

. Prove that

a < 3

and also that

b < 4

and hence that there is only one set of values which satisfy this equation.

Find the six sets of values which satisfy the equation

$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} = 1$

where $a$ , $b$ , $c$ and $d$ are natural numbers and $0 < a < b < c < d$ .