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

ax = 1

then we call this solution the inverse (or reciprocal) of

a

and write it as

a^{- 1}

\frac{1}{a}

. For example the fraction one half in arithmetic modulo 7 is the inverse of 2, that is the solution of

2 x = 1 mod 7

, namely the number 4 because

2 \times 4 = 1 mod 7

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

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

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