1. Write down the addition and multiplication table for Z/5Z.All classes should be written in terms of their canonicalrepresentative (unique representative between 0 and 4).

2. Suppose a ? a' mod n and b ? b' mod n.

(a) Show that a + b ? a' + b' mod n.

(b) Show that a · b ? a' · b' mod n. (An important consequenceof this exercise is that addition and multiplication define mapsZ/nZ × Z/nZ ? Z/nZ. This is not obvious from the definition ofaddition and multiplication, because the definitions require you tochoose representatives for equivalence classes. Different choicesof representatives could conceivably result in different outcomesfor the sum or product of the same pair of elements. A map,however, is not allowed to send the same pair of elements tomultiple distinct elements. For example, in Z/123Z, we have [10] +[5] = [10 + 5] = [15] by definition. However, [10] = [625] and [5]= [866]. Applying the definition of sum again yields [625] + [866]= [1491]. If addition does define a map, then it should be the casethat [15]=[1491], otherwise the pair ([10],[5]) would be mapped totwo distinct values. A simple check shows that this is indeed thecase here, and this exercise shows that this will be true ingeneral.)

3. We extend the notion of divisibility to Z/nZ in the obviousway: [a]|[b] if [b] = [k] · [a] for some k ? Z.

(a) Prove that [a]|[b] if and only if gcd(a, n)|b. (Hint: useproblem 1: Let a, b, c ? Z. Prove that c = ma + nb for some m, n ?Z if and only if gcd(a, b)|c.)

(b) Conclude from part (a) that [a] has a multiplicative inverseif and only if gcd(a, n) = 1.