In arithmetic modulo 7 ( $Z_7$) one integer is equal to another if the difference between the two integers is a multiple of 7. Rather like the days of the week, in $Z_7$ we only need seven numbers and they are usually named 0, 1, 2, 3, 4, 5 and 6. If there is a solution in $Z_7$ to the equation $\mathrm{ax}=1$ then we call this solution the inverse (or reciprocal) of $a$ and write it as $a^{-1}$ or $\frac{1}{a}$. For example the fraction one half in arithmetic modulo 7 is the inverse of 2, that is the solution of $2x=1mod7$, namely the number 4 because $2\times 4=1mod7$. What are the fractions one third, one quarter, one fifth and one sixth in arithmetic modulo 7? Explain why all fractions in arithmetic modulo 7 are equivalent to one of the following set of numbers $\{0,1,2,3,4,5,6\}$. Show that in $Z_7$ there are six different solutions to the equation

${\displaystyle \frac{1}{x}+\frac{1}{y}=\frac{1}{x+y}.}$

Show that, by way of contrast, when working with real numbers this equation has no real solutions.