 |
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 $a^{-1}$ or ${1\over a}$. For example the fraction one
half in arithmetic modulo 7 is the inverse of 2, that is the
solution of $2x=1 \pmod 7$, namely the number 4 because
$2\times 4 = 1 \pmod 7$.
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
$${1\over x} + {1\over y} = {1\over {x+y}}.$$
Show that, by way of contrast, when working with real numbers
this equation has no real solutions.