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

1 x + 1 y = 1 x+y .

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