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 36 is divided by 8.
36 · 8 · Modular arithmetic & remai…
2. Find the remainder when 65 is divided by 11.
65 · 11 · Modular arithmetic & remai…
3. Find the remainder when 35 is divided by 8.
35 · 8 · Modular arithmetic & remai…
4. Find the remainder when 33 is divided by 9.
33 · 9 · Modular arithmetic & remai…
5. Find the remainder when 189 is divided by 11.
189 · 11 · Modular arithmetic & remai…
6. Find the remainder when 80 is divided by 9.
80 · 9 · Modular arithmetic & remai…
7. Find the remainder when 59 is divided by 6.
59 · 6 · Modular arithmetic & remai…
8. Find the remainder when 227 is divided by 12.
227 · 12 · Modular arithmetic & remai…
9. Find the remainder when 158 is divided by 9.
158 · 9 · Modular arithmetic & remai…
10. Find the remainder when 55 is divided by 9.
55 · 9 · Modular arithmetic & remai…
11. Find the remainder when 49 is divided by 9.
49 · 9 · Modular arithmetic & remai…
12. Find the remainder when 31 is divided by 6.
31 · 6 · Modular arithmetic & remai…

12 questions · Seed 43 · 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 36 is divided by 8.
4
36 = 8 × 4 + 4, so the remainder is 4.
2. Find the remainder when 65 is divided by 11.
10
65 = 11 × 5 + 10, so the remainder is 10.
3. Find the remainder when 35 is divided by 8.
3
35 = 8 × 4 + 3, so the remainder is 3.
4. Find the remainder when 33 is divided by 9.
6
33 = 9 × 3 + 6, so the remainder is 6.
5. Find the remainder when 189 is divided by 11.
2
189 = 11 × 17 + 2, so the remainder is 2.
6. Find the remainder when 80 is divided by 9.
8
80 = 9 × 8 + 8, so the remainder is 8.
7. Find the remainder when 59 is divided by 6.
5
59 = 6 × 9 + 5, so the remainder is 5.
8. Find the remainder when 227 is divided by 12.
11
227 = 12 × 18 + 11, so the remainder is 11.
9. Find the remainder when 158 is divided by 9.
5
158 = 9 × 17 + 5, so the remainder is 5.
10. Find the remainder when 55 is divided by 9.
1
55 = 9 × 6 + 1, so the remainder is 1.
11. Find the remainder when 49 is divided by 9.
4
49 = 9 × 5 + 4, so the remainder is 4.
12. Find the remainder when 31 is divided by 6.
1
31 = 6 × 5 + 1, so the remainder is 1.

12 questions · Seed 43 · Age guidance, not curriculum certification

Worksheet StudioPractice pack · Ages 16-18

Take your time. Show your thinking.

1. Find the remainder when 36 is divided by 8.
36 · 8 · Modular arithmetic & remai…
2. Find the remainder when 65 is divided by 11.
65 · 11 · Modular arithmetic & remai…
3. Find the remainder when 35 is divided by 8.
35 · 8 · Modular arithmetic & remai…
4. Find the remainder when 33 is divided by 9.
33 · 9 · Modular arithmetic & remai…
5. Find the remainder when 189 is divided by 11.
189 · 11 · Modular arithmetic & remai…
6. Find the remainder when 80 is divided by 9.
80 · 9 · Modular arithmetic & remai…
7. Find the remainder when 59 is divided by 6.
59 · 6 · Modular arithmetic & remai…
8. Find the remainder when 227 is divided by 12.
227 · 12 · Modular arithmetic & remai…
9. Find the remainder when 158 is divided by 9.
158 · 9 · Modular arithmetic & remai…
10. Find the remainder when 55 is divided by 9.
55 · 9 · Modular arithmetic & remai…
11. Find the remainder when 49 is divided by 9.
49 · 9 · Modular arithmetic & remai…
12. Find the remainder when 31 is divided by 6.
31 · 6 · Modular arithmetic & remai…

12 questions · Seed 43 · Age guidance, not curriculum certification

Worksheet StudioAnswer key · Ages 16-18

Answers and reasoning. Review before teaching.

1. Find the remainder when 36 is divided by 8.
4
36 = 8 × 4 + 4, so the remainder is 4.
2. Find the remainder when 65 is divided by 11.
10
65 = 11 × 5 + 10, so the remainder is 10.
3. Find the remainder when 35 is divided by 8.
3
35 = 8 × 4 + 3, so the remainder is 3.
4. Find the remainder when 33 is divided by 9.
6
33 = 9 × 3 + 6, so the remainder is 6.
5. Find the remainder when 189 is divided by 11.
2
189 = 11 × 17 + 2, so the remainder is 2.
6. Find the remainder when 80 is divided by 9.
8
80 = 9 × 8 + 8, so the remainder is 8.
7. Find the remainder when 59 is divided by 6.
5
59 = 6 × 9 + 5, so the remainder is 5.
8. Find the remainder when 227 is divided by 12.
11
227 = 12 × 18 + 11, so the remainder is 11.
9. Find the remainder when 158 is divided by 9.
5
158 = 9 × 17 + 5, so the remainder is 5.
10. Find the remainder when 55 is divided by 9.
1
55 = 9 × 6 + 1, so the remainder is 1.
11. Find the remainder when 49 is divided by 9.
4
49 = 9 × 5 + 4, so the remainder is 4.
12. Find the remainder when 31 is divided by 6.
1
31 = 6 × 5 + 1, so the remainder is 1.

12 questions · Seed 43 · Age guidance, not curriculum certification