
This set has eight elements:
ℤ
+
9
= f½1, ½2, ½3, ½4, ½5, ½6, ½7, ½8g:
If we want to avoid calculating all 64 entries for the operation table, we can try to think where pro-
blems could arise. Multiplying by the identity is never a cause of trouble, so w e can skip multiplication
by [1]. Let’s what happens when we multiply by [2]:
½2×
9
½2= ½4½2×
9
½3= ½6½2×
9
½4= ½8½2×
9
½5= ½1
½2×
9
½6= ½3½2×
9
½7= ½5½2×
9
½8= ½7:
Everything seems to work just fine. What about using [3]?
½2×
9
½3= ½6½3×
9
½3= ½0½3×
9
½6= ½0
We might as well stop here. The operation ×
9
is not closed on ℤ
+
9
. Every time that we obtain nine or
one of its multiples as a product, ...