Egyptian fraction? What's that?

Number Theory Level 4

The Egyptian fraction of $\frac 57$ can be expressed as $\frac 1a + \frac 1b + \frac 1c$

Evaluate the maximum value of $a+b+c$

1 solution

A Former Brilliant Member
Nov 24, 2018

The greatest fraction of form $\dfrac 1p < \dfrac 57$ is $\dfrac 12.$

We substract $\dfrac 57 - \dfrac 12 = \dfrac {3}{14}.$

The greatest fraction of form $\dfrac 1p < \dfrac {3}{14}$ is $\dfrac 15 .$

We substract $\dfrac {3}{14} - \dfrac 15 = \dfrac {1}{70} .$

Therefore, $\dfrac 57 = \dfrac 1a + \dfrac 1b + \dfrac 1c = \dfrac 12 + \dfrac 15 + \dfrac {1}{70} .$

Hence, $\boxed{a + b + c = 2 + 5 + 70 = {\color{#20A900}{77}}} .$