ReasoningAges 13-15

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 13-15

Take your time. Show your thinking.

1. Find the remainder when 77 is divided by 5.
77 · 5 · Modular arithmetic & remai…
2. Find the remainder when 64 is divided by 7.
64 · 7 · Modular arithmetic & remai…
3. Find the remainder when 107 is divided by 6.
107 · 6 · Modular arithmetic & remai…
4. Find the remainder when 219 is divided by 12.
219 · 12 · Modular arithmetic & remai…
5. Find the remainder when 65 is divided by 6.
65 · 6 · Modular arithmetic & remai…
6. Find the remainder when 81 is divided by 7.
81 · 7 · Modular arithmetic & remai…
7. Find the remainder when 163 is divided by 11.
163 · 11 · Modular arithmetic & remai…
8. Find the remainder when 89 is divided by 7.
89 · 7 · Modular arithmetic & remai…
9. Find the remainder when 57 is divided by 5.
57 · 5 · Modular arithmetic & remai…
10. Find the remainder when 192 is divided by 11.
192 · 11 · Modular arithmetic & remai…
11. Find the remainder when 105 is divided by 6.
105 · 6 · Modular arithmetic & remai…
12. Find the remainder when 109 is divided by 8.
109 · 8 · Modular arithmetic & remai…

12 questions · Seed 42 · Age guidance, not curriculum certification

View the answer key

Worksheet StudioAnswer key · Ages 13-15

Answers and reasoning. Review before teaching.

1. Find the remainder when 77 is divided by 5.
2
77 = 5 × 15 + 2, so the remainder is 2.
2. Find the remainder when 64 is divided by 7.
1
64 = 7 × 9 + 1, so the remainder is 1.
3. Find the remainder when 107 is divided by 6.
5
107 = 6 × 17 + 5, so the remainder is 5.
4. Find the remainder when 219 is divided by 12.
3
219 = 12 × 18 + 3, so the remainder is 3.
5. Find the remainder when 65 is divided by 6.
5
65 = 6 × 10 + 5, so the remainder is 5.
6. Find the remainder when 81 is divided by 7.
4
81 = 7 × 11 + 4, so the remainder is 4.
7. Find the remainder when 163 is divided by 11.
9
163 = 11 × 14 + 9, so the remainder is 9.
8. Find the remainder when 89 is divided by 7.
5
89 = 7 × 12 + 5, so the remainder is 5.
9. Find the remainder when 57 is divided by 5.
2
57 = 5 × 11 + 2, so the remainder is 2.
10. Find the remainder when 192 is divided by 11.
5
192 = 11 × 17 + 5, so the remainder is 5.
11. Find the remainder when 105 is divided by 6.
3
105 = 6 × 17 + 3, so the remainder is 3.
12. Find the remainder when 109 is divided by 8.
5
109 = 8 × 13 + 5, so the remainder is 5.

12 questions · Seed 42 · Age guidance, not curriculum certification

Worksheet StudioPractice pack · Ages 13-15

Take your time. Show your thinking.

1. Find the remainder when 77 is divided by 5.
77 · 5 · Modular arithmetic & remai…
2. Find the remainder when 64 is divided by 7.
64 · 7 · Modular arithmetic & remai…
3. Find the remainder when 107 is divided by 6.
107 · 6 · Modular arithmetic & remai…
4. Find the remainder when 219 is divided by 12.
219 · 12 · Modular arithmetic & remai…
5. Find the remainder when 65 is divided by 6.
65 · 6 · Modular arithmetic & remai…
6. Find the remainder when 81 is divided by 7.
81 · 7 · Modular arithmetic & remai…
7. Find the remainder when 163 is divided by 11.
163 · 11 · Modular arithmetic & remai…
8. Find the remainder when 89 is divided by 7.
89 · 7 · Modular arithmetic & remai…
9. Find the remainder when 57 is divided by 5.
57 · 5 · Modular arithmetic & remai…
10. Find the remainder when 192 is divided by 11.
192 · 11 · Modular arithmetic & remai…
11. Find the remainder when 105 is divided by 6.
105 · 6 · Modular arithmetic & remai…
12. Find the remainder when 109 is divided by 8.
109 · 8 · Modular arithmetic & remai…

12 questions · Seed 42 · Age guidance, not curriculum certification

Worksheet StudioAnswer key · Ages 13-15

Answers and reasoning. Review before teaching.

1. Find the remainder when 77 is divided by 5.
2
77 = 5 × 15 + 2, so the remainder is 2.
2. Find the remainder when 64 is divided by 7.
1
64 = 7 × 9 + 1, so the remainder is 1.
3. Find the remainder when 107 is divided by 6.
5
107 = 6 × 17 + 5, so the remainder is 5.
4. Find the remainder when 219 is divided by 12.
3
219 = 12 × 18 + 3, so the remainder is 3.
5. Find the remainder when 65 is divided by 6.
5
65 = 6 × 10 + 5, so the remainder is 5.
6. Find the remainder when 81 is divided by 7.
4
81 = 7 × 11 + 4, so the remainder is 4.
7. Find the remainder when 163 is divided by 11.
9
163 = 11 × 14 + 9, so the remainder is 9.
8. Find the remainder when 89 is divided by 7.
5
89 = 7 × 12 + 5, so the remainder is 5.
9. Find the remainder when 57 is divided by 5.
2
57 = 5 × 11 + 2, so the remainder is 2.
10. Find the remainder when 192 is divided by 11.
5
192 = 11 × 17 + 5, so the remainder is 5.
11. Find the remainder when 105 is divided by 6.
3
105 = 6 × 17 + 3, so the remainder is 3.
12. Find the remainder when 109 is divided by 8.
5
109 = 8 × 13 + 5, so the remainder is 5.

12 questions · Seed 42 · Age guidance, not curriculum certification