ABSTRACT
In this chapter we analyze how multiplication works in the integers Z, and in particular when division is possible. This is more interesting than asking how multiplication works in the rational numbers Q, where division is always possible (except for division by zero). We all learned at a very young age that we can always divide one
integer by another non-zero integer, as long as we allow for a remainder. For example, 326÷21 gives quotient 15 with remainder 11. The actual computation used to produce this result is our usual long division. Note that the division process halts when we arrive at a number less than the divisor. In this case 11 is less than 21, and so our division process stops. We can record the result of this calculation succinctly as
326 = (21)(15) + 11, where 0 ≤ 11 < 21.
