Apply Modular Multiplication: Clocks, Cycles, and Scheduling

1) A digital clock shows 3:00. After multiplying 3 hours × 4 alarms (mod 12), what time will the alarms overlap?

10:00

0:00 (12:00)

6:00

9:00

3 × 4 = 12; 12 mod 12 = 0 → 12:00.

Explanation

3 × 4 = 12; 12 mod 12 = 0 → 12:00.

2) A warehouse packs boxes of 8 items. If 7 workers each fill 9 boxes, how many items remain if counted mod 8?

1

7

0

2

7 × 9 = 63 boxes, each with 8 items → 63×8 is a multiple of 8, remainder 0.

Explanation

7 × 9 = 63 boxes, each with 8 items → 63×8 is a multiple of 8, remainder 0.

3) A baker bakes trays of 12 pastries. If 25 batches of 8 pastries are made (200 pastries total), how many pastries are left over after filling complete trays of 12?

2

4

8

10

200 ÷ 12 = 16 trays (192 pastries), remainder 8.

Explanation

200 ÷ 12 = 16 trays (192 pastries), remainder 8.

4) A classroom rotates 5 science kits among 8 groups. After 7 rotations, each kit moves 5 groups forward each time. What is each kit’s final position offset from its starting group, modulo 8?

1

2

3

7

7 × 5 = 35; 35 mod 8 = 3.

Explanation

7 × 5 = 35; 35 mod 8 = 3.

5) A rocket system checks every 20 seconds and another system checks every 18 seconds. If multiplied cycles are considered (mod 60), when will both check align?

0 (start)

18

36

54

20 × 18 = 360; 360 mod 60 = 0 → both align at the start.

Explanation

20 × 18 = 360; 360 mod 60 = 0 → both align at the start.