Question

Do the following calculations in \(\mathbb{Z}_{9}\), in each case expressing your answer as \([a]\) with \(0 \leq a \leq 8\) (a) \([8]+[8]\) (b) \([24]+[11]\) (c) [21]\(\cdot[15]\) (d) [8]\(\cdot[8]\)

Short answer

\([8]+[8] = [7]\), \([24]+[11] = [8]\), \([21]\cdot[15] = [6]\), \([8]\cdot[8] = [1]\).

Step-by-step solution

Compute [8]+[8]

To compute \([8]+[8]\), add the representative numbers, 8 and 8, to get 16. Then, determine the remainder when 16 is divided by 9, which is 7. Thus, \([8]+[8] = [7]\).

Compute [24]+[11]

To compute \([24]+[11]\), add the representative numbers, 24 and 11, to get 35. Then, determine the remainder when 35 is divided by 9, which is 8. Thus, \([24]+[11] = [8]\).

Compute [21]·[15]

To compute \([21]\cdot[15]\), multiply the representative numbers, 21 and 15, to get 315. Then, determine the remainder when 315 is divided by 9, which is 6. Thus, \([21]\cdot[15] = [6]\).

Compute [8]·[8]

To compute \([8]\cdot[8]\), multiply the representative numbers, 8 and 8, to get 64. Then, determine the remainder when 64 is divided by 9, which is 1. Thus, \([8]\cdot[8] = [1]\).