Clock Arithmetic Quiz: Clock Arithmetic With Remainders

1) On a 12-hour clock, 15 is congruent to 3 mod 12.

True

False

The answer is True. 15 minus 3 = 12, which is exactly one multiple of 12. Two numbers are congruent mod 12 when their difference is a multiple of 12. Since 12 is a multiple of 12, 15 and 3 represent the same position on a 12-hour clock.

Explanation

The answer is True. 15 minus 3 = 12, which is exactly one multiple of 12. Two numbers are congruent mod 12 when their difference is a multiple of 12. Since 12 is a multiple of 12, 15 and 3 represent the same position on a 12-hour clock.

2) On a 24-hour clock, 06:00 minus 19 hours = what time?

9

10

11

12

6 minus 19 = -13. Convert to positive mod 24: -13 + 24 = 11. The clock reads 11:00. Option A gives 9, option B gives 10, option D gives 12, none of which correctly add 24 to bring -13 into the valid range.

Explanation

6 minus 19 = -13. Convert to positive mod 24: -13 + 24 = 11. The clock reads 11:00. Option A gives 9, option B gives 10, option D gives 12, none of which correctly add 24 to bring -13 into the valid range.

3) On a 12-hour clock, 13:00 shows as 1:00.

True

False

The answer is True. 13 mod 12 = 1 since 13 minus 12 = 1. A 12-hour clock cycles every 12 hours, so 13 hours after midnight is the same position as 1 hour after midnight. This is why 1pm on a 12-hour clock displays as 1:00, the same as 1am.

Explanation

The answer is True. 13 mod 12 = 1 since 13 minus 12 = 1. A 12-hour clock cycles every 12 hours, so 13 hours after midnight is the same position as 1 hour after midnight. This is why 1pm on a 12-hour clock displays as 1:00, the same as 1am.

4) On a 12-hour clock, 8 + 20 hours equals 4.

True

False

The answer is True. 8 + 20 = 28. Find remainder: 28 mod 12 = 4 since 12 times 2 = 24 and 28 minus 24 = 4. The clock shows 4. This confirms that starting at 8 and advancing 20 hours brings you to 4 on a 12-hour clock.

Explanation

The answer is True. 8 + 20 = 28. Find remainder: 28 mod 12 = 4 since 12 times 2 = 24 and 28 minus 24 = 4. The clock shows 4. This confirms that starting at 8 and advancing 20 hours brings you to 4 on a 12-hour clock.

5) On a 12-hour clock, adding 12 hours leaves the same time.

True

False

The answer is True. 12 mod 12 = 0, so adding 12 hours is the same as adding 0 hours on the clock face. The clock completes exactly one full cycle and returns to the same position. For example, 3 o'clock plus 12 hours is still 3 o'clock.

Explanation

The answer is True. 12 mod 12 = 0, so adding 12 hours is the same as adding 0 hours on the clock face. The clock completes exactly one full cycle and returns to the same position. For example, 3 o'clock plus 12 hours is still 3 o'clock.

6) On a 12-hour clock, what time is 11 + 15 hours?

1

2

3

4

11 + 15 = 26. Find remainder: 26 mod 12 = 2 since 12 times 2 = 24 and 26 minus 24 = 2. The clock shows 2. Option A gives 1, option C gives 3, option D gives 4, none of which equal 26 mod 12.

Explanation

11 + 15 = 26. Find remainder: 26 mod 12 = 2 since 12 times 2 = 24 and 26 minus 24 = 2. The clock shows 2. Option A gives 1, option C gives 3, option D gives 4, none of which equal 26 mod 12.

7) On a 24-hour clock, 24 is congruent to 0 mod 24.

True

False

The answer is True. 24 minus 0 = 24, which is exactly one multiple of 24. A full day of 24 hours brings the clock back to the same position as 0, which is midnight. This is why midnight on a 24-hour clock is written as 00:00.

Explanation

The answer is True. 24 minus 0 = 24, which is exactly one multiple of 24. A full day of 24 hours brings the clock back to the same position as 0, which is midnight. This is why midnight on a 24-hour clock is written as 00:00.

8) On a 12-hour clock, starting at 3 and adding 50 hours gives what time?

4

5

6

8

3 + 50 = 53. Find remainder: 53 mod 12 = 5 since 12 times 4 = 48 and 53 minus 48 = 5. The clock shows 5. Option A gives 4, option C gives 6, option D gives 8, none of which equal 53 mod 12.

Explanation

3 + 50 = 53. Find remainder: 53 mod 12 = 5 since 12 times 4 = 48 and 53 minus 48 = 5. The clock shows 5. Option A gives 4, option C gives 6, option D gives 8, none of which equal 53 mod 12.

9) Select all numbers congruent to 4 mod 12.

16

28

40

3

16 mod 12 = 4 since 16 minus 12 = 4, confirming A. 28 mod 12 = 4 since 28 minus 24 = 4, confirming B. 40 mod 12 = 4 since 40 minus 36 = 4, confirming C. Option D gives 3, and 3 mod 12 = 3, not 4, so it is not congruent to 4 mod 12.

Explanation

16 mod 12 = 4 since 16 minus 12 = 4, confirming A. 28 mod 12 = 4 since 28 minus 24 = 4, confirming B. 40 mod 12 = 4 since 40 minus 36 = 4, confirming C. Option D gives 3, and 3 mod 12 = 3, not 4, so it is not congruent to 4 mod 12.

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10) On a 24-hour clock starting at 00:00, after 73 hours the clock reads what time?

0

1

2

3

73 mod 24 = 1 since 24 times 3 = 72 and 73 minus 72 = 1. The clock reads 01:00. Option A gives 0, option C gives 2, option D gives 3, none of which equal 73 mod 24.

Explanation

73 mod 24 = 1 since 24 times 3 = 72 and 73 minus 72 = 1. The clock reads 01:00. Option A gives 0, option C gives 2, option D gives 3, none of which equal 73 mod 24.

11) Which congruence is correct modulo 12?

7 is congruent to -5 mod 12

7 is congruent to 5 mod 12

7 is congruent to -7 mod 12

7 is congruent to 1 mod 6

Negative 5 plus 12 = 7, so -5 and 7 differ by exactly 12, confirming they are congruent mod 12. Option B: 7 minus 5 = 2, not a multiple of 12. Option C: 7 minus (-7) = 14, not a multiple of 12. Option D changes the modulus to 6 rather than 12, and 7 mod 6 = 1, which is a different congruence class.

Explanation

Negative 5 plus 12 = 7, so -5 and 7 differ by exactly 12, confirming they are congruent mod 12. Option B: 7 minus 5 = 2, not a multiple of 12. Option C: 7 minus (-7) = 14, not a multiple of 12. Option D changes the modulus to 6 rather than 12, and 7 mod 6 = 1, which is a different congruence class.

12) Select all pairs (a, b) with a congruent to b mod 12.

25 and 1

14 and 2

18 and 7

-1 and 11

25 mod 12 = 1 and 1 mod 12 = 1, confirming A. 14 mod 12 = 2 and 2 mod 12 = 2, confirming B. -1 mod 12 = 11 since -1 + 12 = 11, and 11 mod 12 = 11, confirming D. Option C: 18 mod 12 = 6 but 7 mod 12 = 7, these are not equal so the pair is not congruent mod 12.

Explanation

25 mod 12 = 1 and 1 mod 12 = 1, confirming A. 14 mod 12 = 2 and 2 mod 12 = 2, confirming B. -1 mod 12 = 11 since -1 + 12 = 11, and 11 mod 12 = 11, confirming D. Option C: 18 mod 12 = 6 but 7 mod 12 = 7, these are not equal so the pair is not congruent mod 12.

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13) On a 24-hour clock, 22:00 + 7 hours = what time?

3

4

6

5

22 + 7 = 29. Find remainder: 29 mod 24 = 5 since 29 minus 24 = 5. The clock reads 05:00. Option A gives 3, option B gives 4, option C gives 6, none of which equal 29 mod 24.

Explanation

22 + 7 = 29. Find remainder: 29 mod 24 = 5 since 29 minus 24 = 5. The clock reads 05:00. Option A gives 3, option B gives 4, option C gives 6, none of which equal 29 mod 24.

14) On a 24-hour clock, 5 minus 13 hours = what time?

13

14

15

16

5 minus 13 = -8. Convert to positive mod 24: -8 + 24 = 16. The clock reads 16:00. Option A gives 13, option B gives 14, option C gives 15, none of which correctly add 24 to bring -8 into the valid 0 to 23 range.

Explanation

5 minus 13 = -8. Convert to positive mod 24: -8 + 24 = 16. The clock reads 16:00. Option A gives 13, option B gives 14, option C gives 15, none of which correctly add 24 to bring -8 into the valid 0 to 23 range.

15) On a 12-hour clock, 10 + 35 hours = what time?

7

8

9

10

10 + 35 = 45. Find remainder: 45 mod 12 = 9 since 12 times 3 = 36 and 45 minus 36 = 9. The clock shows 9. Option A gives 7, option B gives 8, option D gives 10, none of which equal 45 mod 12.

Explanation

10 + 35 = 45. Find remainder: 45 mod 12 = 9 since 12 times 3 = 36 and 45 minus 36 = 9. The clock shows 9. Option A gives 7, option B gives 8, option D gives 10, none of which equal 45 mod 12.

16) On a 12-hour clock, what time is 9 + 5 hours?

1

2

3

11

Compute 9 + 5 = 14. On a 12-hour clock, find the remainder when dividing by 12: 14 mod 12 = 2. The clock shows 2. Option A gives 1, option C gives 3, option D gives 11, none of which equal 14 mod 12.

Explanation

Compute 9 + 5 = 14. On a 12-hour clock, find the remainder when dividing by 12: 14 mod 12 = 2. The clock shows 2. Option A gives 1, option C gives 3, option D gives 11, none of which equal 14 mod 12.

17) Modulo 24, which numbers are congruent to -1?

23

-25

47

1

-1 mod 24 = 23 since -1 + 24 = 23, confirming A. -25 + 24 = -1, so -25 is congruent to -1 mod 24, confirming B. 47 = 24 times 1 + 23 = 24 + 23, so 47 mod 24 = 23 which equals -1 mod 24, confirming C. Option D: 1 mod 24 = 1, not 23, so 1 is not congruent to -1 mod 24.

Explanation

-1 mod 24 = 23 since -1 + 24 = 23, confirming A. -25 + 24 = -1, so -25 is congruent to -1 mod 24, confirming B. 47 = 24 times 1 + 23 = 24 + 23, so 47 mod 24 = 23 which equals -1 mod 24, confirming C. Option D: 1 mod 24 = 1, not 23, so 1 is not congruent to -1 mod 24.

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18) On a 12-hour clock, what time is 7 + 19 hours?

1

2

3

4

7 + 19 = 26. Find remainder: 26 mod 12 = 2 since 12 times 2 = 24 and 26 minus 24 = 2. The clock shows 2. Option A gives 1, option C gives 3, option D gives 4, none of which equal 26 mod 12.

Explanation

7 + 19 = 26. Find remainder: 26 mod 12 = 2 since 12 times 2 = 24 and 26 minus 24 = 2. The clock shows 2. Option A gives 1, option C gives 3, option D gives 4, none of which equal 26 mod 12.

19) On a 12-hour clock starting at 12, after 115 hours it will read what time?

5

6

7

8

Treat 12 as 0 on the clock face. 115 mod 12 = 7 since 12 times 9 = 108 and 115 minus 108 = 7. The display shows 7. Option A gives 5, option B gives 6, option D gives 8, none of which equal 115 mod 12.

Explanation

Treat 12 as 0 on the clock face. 115 mod 12 = 7 since 12 times 9 = 108 and 115 minus 108 = 7. The display shows 7. Option A gives 5, option B gives 6, option D gives 8, none of which equal 115 mod 12.

20) On a 24-hour clock, which totals are congruent to 18 mod 24?

18

42

66

100

18 mod 24 = 18 confirming A. 42 = 18 + 24, so 42 mod 24 = 18, confirming B. 66 = 18 + 48 = 18 + 24 times 2, so 66 mod 24 = 18, confirming C. Option D: 100 mod 24 = 4 since 24 times 4 = 96 and 100 minus 96 = 4, which is not 18.

Explanation

18 mod 24 = 18 confirming A. 42 = 18 + 24, so 42 mod 24 = 18, confirming B. 66 = 18 + 48 = 18 + 24 times 2, so 66 mod 24 = 18, confirming C. Option D: 100 mod 24 = 4 since 24 times 4 = 96 and 100 minus 96 = 4, which is not 18.

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