A positive rational number $q$ is expressed in friendly form if it is written as a finite sum of reciprocals of distinct positive integers. For example, $\frac { 4 } { 5 } = \frac { 1 } { 2 } + \frac { 1 } { 4 } + \frac { 1 } { 20 }$. (i) Express the following numbers in friendly form: $\frac { 2 } { 3 } , \frac { 2 } { 5 } , \frac { 23 } { 40 }$. (ii) Let $q$ be a rational number with $0 < q < 1$, and $m$ be the smallest natural number such than $\frac { 1 } { m } \leqslant q$. Suppose $q = \frac { a } { b }$ and $q - \frac { 1 } { m } = \frac { c } { d }$ in their lowest terms. Show that $c < a$. (iii) Suggest a procedure by which any rational $q$ with $0 < q < 1$ can be expressed in friendly form. Use the result in part (ii) to show that the procedure always works, generating distinct reciprocals and finishing within a finite time. (iv) Demonstrate your procedure by finding a friendly form for $\frac { 4 } { 13 }$. (v) Assuming that $\sum _ { n = 1 } ^ { N } \frac { 1 } { n }$ increases without bound as $N$ becomes large, show that every positive rational number can be expressed in friendly form.
\section*{6. For APPLICANTS IN $\left\{ \begin{array} { l } \text { COMPUTER SCIENCE } \\ \text { MATHEMATICS \& COMPUTER SCIENCE } \\ \text { COMPUTER SCIENCE \& PHILOSOPHY } \end{array} \right\}$ ONLY.}
A positive rational number $q$ is expressed in friendly form if it is written as a finite sum of reciprocals of distinct positive integers. For example, $\frac { 4 } { 5 } = \frac { 1 } { 2 } + \frac { 1 } { 4 } + \frac { 1 } { 20 }$.\\
(i) Express the following numbers in friendly form: $\frac { 2 } { 3 } , \frac { 2 } { 5 } , \frac { 23 } { 40 }$.\\
(ii) Let $q$ be a rational number with $0 < q < 1$, and $m$ be the smallest natural number such than $\frac { 1 } { m } \leqslant q$. Suppose $q = \frac { a } { b }$ and $q - \frac { 1 } { m } = \frac { c } { d }$ in their lowest terms. Show that $c < a$.\\
(iii) Suggest a procedure by which any rational $q$ with $0 < q < 1$ can be expressed in friendly form. Use the result in part (ii) to show that the procedure always works, generating distinct reciprocals and finishing within a finite time.\\
(iv) Demonstrate your procedure by finding a friendly form for $\frac { 4 } { 13 }$.\\
(v) Assuming that $\sum _ { n = 1 } ^ { N } \frac { 1 } { n }$ increases without bound as $N$ becomes large, show that every positive rational number can be expressed in friendly form.