ReasoningAges 16-18

Modular arithmetic & remainder cycles

Use division remainders (modulo arithmetic) to predict where a repeating cycle lands. Review suitability for your learners before teaching.

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Worksheet StudioPractice pack · Ages 16-18

Take your time. Show your thinking.

1. Find the remainder when 62 is divided by 12.
62 · 12 · Modular arithmetic & remai…
2. Find the remainder when 26 is divided by 8.
26 · 8 · Modular arithmetic & remai…
3. Find the remainder when 63 is divided by 5.
63 · 5 · Modular arithmetic & remai…
4. Find the remainder when 106 is divided by 12.
106 · 12 · Modular arithmetic & remai…
5. Find the remainder when 76 is divided by 7.
76 · 7 · Modular arithmetic & remai…
6. Find the remainder when 128 is divided by 12.
128 · 12 · Modular arithmetic & remai…
7. Find the remainder when 57 is divided by 9.
57 · 9 · Modular arithmetic & remai…
8. Find the remainder when 38 is divided by 11.
38 · 11 · Modular arithmetic & remai…
9. Find the remainder when 153 is divided by 11.
153 · 11 · Modular arithmetic & remai…
10. Find the remainder when 69 is divided by 8.
69 · 8 · Modular arithmetic & remai…
11. Find the remainder when 41 is divided by 7.
41 · 7 · Modular arithmetic & remai…
12. Find the remainder when 78 is divided by 12.
78 · 12 · Modular arithmetic & remai…

12 questions · Seed 104730 · Age guidance, not curriculum certification

View the answer key

Worksheet StudioAnswer key · Ages 16-18

Answers and reasoning. Review before teaching.

1. Find the remainder when 62 is divided by 12.
2
62 = 12 × 5 + 2, so the remainder is 2.
2. Find the remainder when 26 is divided by 8.
2
26 = 8 × 3 + 2, so the remainder is 2.
3. Find the remainder when 63 is divided by 5.
3
63 = 5 × 12 + 3, so the remainder is 3.
4. Find the remainder when 106 is divided by 12.
10
106 = 12 × 8 + 10, so the remainder is 10.
5. Find the remainder when 76 is divided by 7.
6
76 = 7 × 10 + 6, so the remainder is 6.
6. Find the remainder when 128 is divided by 12.
8
128 = 12 × 10 + 8, so the remainder is 8.
7. Find the remainder when 57 is divided by 9.
3
57 = 9 × 6 + 3, so the remainder is 3.
8. Find the remainder when 38 is divided by 11.
5
38 = 11 × 3 + 5, so the remainder is 5.
9. Find the remainder when 153 is divided by 11.
10
153 = 11 × 13 + 10, so the remainder is 10.
10. Find the remainder when 69 is divided by 8.
5
69 = 8 × 8 + 5, so the remainder is 5.
11. Find the remainder when 41 is divided by 7.
6
41 = 7 × 5 + 6, so the remainder is 6.
12. Find the remainder when 78 is divided by 12.
6
78 = 12 × 6 + 6, so the remainder is 6.

12 questions · Seed 104730 · Age guidance, not curriculum certification

Worksheet StudioPractice pack · Ages 16-18

Take your time. Show your thinking.

1. Find the remainder when 62 is divided by 12.
62 · 12 · Modular arithmetic & remai…
2. Find the remainder when 26 is divided by 8.
26 · 8 · Modular arithmetic & remai…
3. Find the remainder when 63 is divided by 5.
63 · 5 · Modular arithmetic & remai…
4. Find the remainder when 106 is divided by 12.
106 · 12 · Modular arithmetic & remai…
5. Find the remainder when 76 is divided by 7.
76 · 7 · Modular arithmetic & remai…
6. Find the remainder when 128 is divided by 12.
128 · 12 · Modular arithmetic & remai…
7. Find the remainder when 57 is divided by 9.
57 · 9 · Modular arithmetic & remai…
8. Find the remainder when 38 is divided by 11.
38 · 11 · Modular arithmetic & remai…
9. Find the remainder when 153 is divided by 11.
153 · 11 · Modular arithmetic & remai…
10. Find the remainder when 69 is divided by 8.
69 · 8 · Modular arithmetic & remai…
11. Find the remainder when 41 is divided by 7.
41 · 7 · Modular arithmetic & remai…
12. Find the remainder when 78 is divided by 12.
78 · 12 · Modular arithmetic & remai…

12 questions · Seed 104730 · Age guidance, not curriculum certification

Worksheet StudioAnswer key · Ages 16-18

Answers and reasoning. Review before teaching.

1. Find the remainder when 62 is divided by 12.
2
62 = 12 × 5 + 2, so the remainder is 2.
2. Find the remainder when 26 is divided by 8.
2
26 = 8 × 3 + 2, so the remainder is 2.
3. Find the remainder when 63 is divided by 5.
3
63 = 5 × 12 + 3, so the remainder is 3.
4. Find the remainder when 106 is divided by 12.
10
106 = 12 × 8 + 10, so the remainder is 10.
5. Find the remainder when 76 is divided by 7.
6
76 = 7 × 10 + 6, so the remainder is 6.
6. Find the remainder when 128 is divided by 12.
8
128 = 12 × 10 + 8, so the remainder is 8.
7. Find the remainder when 57 is divided by 9.
3
57 = 9 × 6 + 3, so the remainder is 3.
8. Find the remainder when 38 is divided by 11.
5
38 = 11 × 3 + 5, so the remainder is 5.
9. Find the remainder when 153 is divided by 11.
10
153 = 11 × 13 + 10, so the remainder is 10.
10. Find the remainder when 69 is divided by 8.
5
69 = 8 × 8 + 5, so the remainder is 5.
11. Find the remainder when 41 is divided by 7.
6
41 = 7 × 5 + 6, so the remainder is 6.
12. Find the remainder when 78 is divided by 12.
6
78 = 12 × 6 + 6, so the remainder is 6.

12 questions · Seed 104730 · Age guidance, not curriculum certification