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 65 is divided by 12.
65 · 12 · Modular arithmetic & remai…
2. Find the remainder when 62 is divided by 5.
62 · 5 · Modular arithmetic & remai…
3. Find the remainder when 83 is divided by 7.
83 · 7 · Modular arithmetic & remai…
4. Find the remainder when 28 is divided by 9.
28 · 9 · Modular arithmetic & remai…
5. Find the remainder when 91 is divided by 9.
91 · 9 · Modular arithmetic & remai…
6. Find the remainder when 91 is divided by 6.
91 · 6 · Modular arithmetic & remai…
7. Find the remainder when 23 is divided by 5.
23 · 5 · Modular arithmetic & remai…
8. Find the remainder when 136 is divided by 12.
136 · 12 · Modular arithmetic & remai…
9. Find the remainder when 51 is divided by 8.
51 · 8 · Modular arithmetic & remai…
10. Find the remainder when 111 is divided by 8.
111 · 8 · Modular arithmetic & remai…
11. Find the remainder when 73 is divided by 5.
73 · 5 · Modular arithmetic & remai…
12. Find the remainder when 161 is divided by 11.
161 · 11 · Modular arithmetic & remai…

12 questions · Seed 44 · 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 65 is divided by 12.
5
65 = 12 × 5 + 5, so the remainder is 5.
2. Find the remainder when 62 is divided by 5.
2
62 = 5 × 12 + 2, so the remainder is 2.
3. Find the remainder when 83 is divided by 7.
6
83 = 7 × 11 + 6, so the remainder is 6.
4. Find the remainder when 28 is divided by 9.
1
28 = 9 × 3 + 1, so the remainder is 1.
5. Find the remainder when 91 is divided by 9.
1
91 = 9 × 10 + 1, so the remainder is 1.
6. Find the remainder when 91 is divided by 6.
1
91 = 6 × 15 + 1, so the remainder is 1.
7. Find the remainder when 23 is divided by 5.
3
23 = 5 × 4 + 3, so the remainder is 3.
8. Find the remainder when 136 is divided by 12.
4
136 = 12 × 11 + 4, so the remainder is 4.
9. Find the remainder when 51 is divided by 8.
3
51 = 8 × 6 + 3, so the remainder is 3.
10. Find the remainder when 111 is divided by 8.
7
111 = 8 × 13 + 7, so the remainder is 7.
11. Find the remainder when 73 is divided by 5.
3
73 = 5 × 14 + 3, so the remainder is 3.
12. Find the remainder when 161 is divided by 11.
7
161 = 11 × 14 + 7, so the remainder is 7.

12 questions · Seed 44 · Age guidance, not curriculum certification

Worksheet StudioPractice pack · Ages 16-18

Take your time. Show your thinking.

1. Find the remainder when 65 is divided by 12.
65 · 12 · Modular arithmetic & remai…
2. Find the remainder when 62 is divided by 5.
62 · 5 · Modular arithmetic & remai…
3. Find the remainder when 83 is divided by 7.
83 · 7 · Modular arithmetic & remai…
4. Find the remainder when 28 is divided by 9.
28 · 9 · Modular arithmetic & remai…
5. Find the remainder when 91 is divided by 9.
91 · 9 · Modular arithmetic & remai…
6. Find the remainder when 91 is divided by 6.
91 · 6 · Modular arithmetic & remai…
7. Find the remainder when 23 is divided by 5.
23 · 5 · Modular arithmetic & remai…
8. Find the remainder when 136 is divided by 12.
136 · 12 · Modular arithmetic & remai…
9. Find the remainder when 51 is divided by 8.
51 · 8 · Modular arithmetic & remai…
10. Find the remainder when 111 is divided by 8.
111 · 8 · Modular arithmetic & remai…
11. Find the remainder when 73 is divided by 5.
73 · 5 · Modular arithmetic & remai…
12. Find the remainder when 161 is divided by 11.
161 · 11 · Modular arithmetic & remai…

12 questions · Seed 44 · Age guidance, not curriculum certification

Worksheet StudioAnswer key · Ages 16-18

Answers and reasoning. Review before teaching.

1. Find the remainder when 65 is divided by 12.
5
65 = 12 × 5 + 5, so the remainder is 5.
2. Find the remainder when 62 is divided by 5.
2
62 = 5 × 12 + 2, so the remainder is 2.
3. Find the remainder when 83 is divided by 7.
6
83 = 7 × 11 + 6, so the remainder is 6.
4. Find the remainder when 28 is divided by 9.
1
28 = 9 × 3 + 1, so the remainder is 1.
5. Find the remainder when 91 is divided by 9.
1
91 = 9 × 10 + 1, so the remainder is 1.
6. Find the remainder when 91 is divided by 6.
1
91 = 6 × 15 + 1, so the remainder is 1.
7. Find the remainder when 23 is divided by 5.
3
23 = 5 × 4 + 3, so the remainder is 3.
8. Find the remainder when 136 is divided by 12.
4
136 = 12 × 11 + 4, so the remainder is 4.
9. Find the remainder when 51 is divided by 8.
3
51 = 8 × 6 + 3, so the remainder is 3.
10. Find the remainder when 111 is divided by 8.
7
111 = 8 × 13 + 7, so the remainder is 7.
11. Find the remainder when 73 is divided by 5.
3
73 = 5 × 14 + 3, so the remainder is 3.
12. Find the remainder when 161 is divided by 11.
7
161 = 11 × 14 + 7, so the remainder is 7.

12 questions · Seed 44 · Age guidance, not curriculum certification