Apply Congruences to Cycles, Calendars, and Repeating Processes

1) A theater rotates 5 shows. If show 1 is today, which show is on the 23rd day?

2

3

4

5

((23 − 1) mod 5) + 1 = (22 mod 5) + 1 = 2 + 1 = 3. Show 3 plays on day 23.

Explanation

((23 − 1) mod 5) + 1 = (22 mod 5) + 1 = 2 + 1 = 3. Show 3 plays on day 23.

2) A bus runs every 18 minutes. If the first bus is at 10:00, when is the 15th bus?

13:20

14:12

13:36

14:00

There are 14 intervals × 18 min = 252 min = 4 h 12 min. 10:00 + 4:12 = 14:12.

Explanation

There are 14 intervals × 18 min = 252 min = 4 h 12 min. 10:00 + 4:12 = 14:12.

3) A clock shows 7:00. What time will it show 137 hours later?

8:00

9:00

12:00

1:00

137 mod 12 = 5. 7 + 5 = 12:00 on the clock.

Explanation

137 mod 12 = 5. 7 + 5 = 12:00 on the clock.

4) A sequence follows n ≡ 2 (mod 5). Which of these is in the sequence?

7

9

14

13

7 mod 5 = 2, so 7 fits the rule.

Explanation

7 mod 5 = 2, so 7 fits the rule.

5) A calendar repeats every 12 months. If an event is in April, when is it again after 50 months?

March

June

July

August

50 mod 12 = 2. Two months after April = June.

Explanation

50 mod 12 = 2. Two months after April = June.

6) A digital clock shows 11:00. What time is it 27 hours later?

1:00

4:00

2:00

3:00

27 mod 12 = 3. Add 3 hours to 11:00 → 2:00.

Explanation

27 mod 12 = 3. Add 3 hours to 11:00 → 2:00.

7) A 7-day schedule repeats. If today is Monday, what day will it be in 50 days?

Tuesday

Thursday

Friday

Saturday

50 mod 7 = 1. One day after Monday → Tuesday.

Explanation

50 mod 7 = 1. One day after Monday → Tuesday.

8) A pizza shop sells slices in packs of 8. If 59 slices are sold, how many are in the last pack?

4

5

3

7

59 mod 8 = 3. Three slices fill the final pack.

Explanation

59 mod 8 = 3. Three slices fill the final pack.

9) A machine beeps every 9 seconds. If it beeps at time 0, when will it next beep at a time ending in 7?

27 s

45 s

54 s

63 s

Solve 9n ≡ 7 (mod 10). n = 3 works. 9 × 3 = 27 s.

Explanation

Solve 9n ≡ 7 (mod 10). n = 3 works. 9 × 3 = 27 s.

10) A runner completes 400 m laps. After 2500 m, what distance into the lap are they?

250 m

200 m

100 m

300 m

2500 mod 400 = 100 → 100 m into the next lap.

Explanation

2500 mod 400 = 100 → 100 m into the next lap.

Applications of Congruence

1) A theater rotates 5 shows. If show 1 is today, which show is on the 23rd day?

2)

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

2) A bus runs every 18 minutes. If the first bus is at 10:00, when is the 15th bus?

3) A clock shows 7:00. What time will it show 137 hours later?

4) A sequence follows n ≡ 2 (mod 5). Which of these is in the sequence?

5) A calendar repeats every 12 months. If an event is in April, when is it again after 50 months?

6) A digital clock shows 11:00. What time is it 27 hours later?

7) A 7-day schedule repeats. If today is Monday, what day will it be in 50 days?

8) A pizza shop sells slices in packs of 8. If 59 slices are sold, how many are in the last pack?

9) A machine beeps every 9 seconds. If it beeps at time 0, when will it next beep at a time ending in 7?

10) A runner completes 400 m laps. After 2500 m, what distance into the lap are they?