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.

6) A spinner game has 9 positions. A player spins 12 × 5 times. Where do they land (mod 9)?

2

3

6

9

12 × 5 = 60; 60 mod 9 = 6.

Explanation

12 × 5 = 60; 60 mod 9 = 6.

7) A festival has 10 booths. Tickets are grouped in packs of 7. If someone buys 13 packs, how many tickets align with booths (mod 10)?

1

3

5

7

13 × 7 = 91; 91 mod 10 = 1.

Explanation

13 × 7 = 91; 91 mod 10 = 1.

8) A code lock resets every 9 turns. If a student enters 14 × 6 turns, where will it stop (mod 9)?

1

2

3

6

14 × 6 = 84; 84 mod 9 = 3.

Explanation

14 × 6 = 84; 84 mod 9 = 3.

9) A Ferris wheel has 16 seats. If 7 people ride 11 times each, which seat number matches their exit seat (mod 16)?

2

13

15

16

7 × 11 = 77; 77 mod 16 = 13.

Explanation

7 × 11 = 77; 77 mod 16 = 13.

10) A chess clock counts moves mod 5. If 18 × 4 moves are made, what is the position on the counter?

0

1

2

4

18 × 4 = 72; 72 mod 5 = 2.

Explanation

18 × 4 = 72; 72 mod 5 = 2.

Real-World Applications of Modular Multiplication

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

2)

What first name or nickname would you like us to use?

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

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?

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?

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?

6) A spinner game has 9 positions. A player spins 12 × 5 times. Where do they land (mod 9)?

7) A festival has 10 booths. Tickets are grouped in packs of 7. If someone buys 13 packs, how many tickets align with booths (mod 10)?

8) A code lock resets every 9 turns. If a student enters 14 × 6 turns, where will it stop (mod 9)?

9) A Ferris wheel has 16 seats. If 7 people ride 11 times each, which seat number matches their exit seat (mod 16)?

10) A chess clock counts moves mod 5. If 18 × 4 moves are made, what is the position on the counter?