EDB — 1Z8

[1Z8] \(ℤ\) are the relative integers with the usual operations. Let \(p≥ 1\) a fixed integer. Consider the equivalence relation

that is, they are equivalent when \(n-m\) is divisible by \(p\).

Show that there are \(p\) equivalence classes \([0],[1],\ldots [p-1]\) We indicate the quotient space with \(ℤ/(pℤ)\) or more briefly \(ℤ_ p\).

Show that the usual operations of sum and product in \(ℤ\) are invariant (in the sense defined in [(2.194)]↺↻), hence they pass to the quotient.