In arithmetic modulo 7 (Z7) 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 Z7 we only need seven
numbers and they are usually named 0, 1, 2, 3, 4,
5 and 6.
If there is a solution in Z7 to the equation
ax=1 then we call this solution the inverse (or
reciprocal) of a and write it as a-1 or
1/a. For example the fraction one half
in arithmetic modulo 7 is the inverse of 2, that
is the solution of 2x=1 mod 7, namely the
number 4 because 2×4 = 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 Z7 there are six different
solutions to the equation
1x
+
1y
=
1x+y
.
Show that, by way of contrast, when working with
real numbers this equation has no real solutions.
