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 54 is divided by 5.
54 · 5 · Modular arithmetic & remai…
2. Find the remainder when 74 is divided by 5.
74 · 5 · Modular arithmetic & remai…
3. Find the remainder when 38 is divided by 5.
38 · 5 · Modular arithmetic & remai…
4. Find the remainder when 61 is divided by 5.
61 · 5 · Modular arithmetic & remai…
5. Find the remainder when 52 is divided by 8.
52 · 8 · Modular arithmetic & remai…
6. Find the remainder when 159 is divided by 12.
159 · 12 · Modular arithmetic & remai…
7. Find the remainder when 87 is divided by 5.
87 · 5 · Modular arithmetic & remai…
8. Find the remainder when 41 is divided by 6.
41 · 6 · Modular arithmetic & remai…
9. Find the remainder when 191 is divided by 12.
191 · 12 · Modular arithmetic & remai…
10. Find the remainder when 63 is divided by 5.
63 · 5 · Modular arithmetic & remai…
11. Find the remainder when 158 is divided by 11.
158 · 11 · Modular arithmetic & remai…
12. Find the remainder when 120 is divided by 11.
120 · 11 · Modular arithmetic & remai…

12 questions · Seed 104736 · 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 54 is divided by 5.
4
54 = 5 × 10 + 4, so the remainder is 4.
2. Find the remainder when 74 is divided by 5.
4
74 = 5 × 14 + 4, so the remainder is 4.
3. Find the remainder when 38 is divided by 5.
3
38 = 5 × 7 + 3, so the remainder is 3.
4. Find the remainder when 61 is divided by 5.
1
61 = 5 × 12 + 1, so the remainder is 1.
5. Find the remainder when 52 is divided by 8.
4
52 = 8 × 6 + 4, so the remainder is 4.
6. Find the remainder when 159 is divided by 12.
3
159 = 12 × 13 + 3, so the remainder is 3.
7. Find the remainder when 87 is divided by 5.
2
87 = 5 × 17 + 2, so the remainder is 2.
8. Find the remainder when 41 is divided by 6.
5
41 = 6 × 6 + 5, so the remainder is 5.
9. Find the remainder when 191 is divided by 12.
11
191 = 12 × 15 + 11, so the remainder is 11.
10. Find the remainder when 63 is divided by 5.
3
63 = 5 × 12 + 3, so the remainder is 3.
11. Find the remainder when 158 is divided by 11.
4
158 = 11 × 14 + 4, so the remainder is 4.
12. Find the remainder when 120 is divided by 11.
10
120 = 11 × 10 + 10, so the remainder is 10.

12 questions · Seed 104736 · Age guidance, not curriculum certification

Worksheet StudioPractice pack · Ages 16-18

Take your time. Show your thinking.

1. Find the remainder when 54 is divided by 5.
54 · 5 · Modular arithmetic & remai…
2. Find the remainder when 74 is divided by 5.
74 · 5 · Modular arithmetic & remai…
3. Find the remainder when 38 is divided by 5.
38 · 5 · Modular arithmetic & remai…
4. Find the remainder when 61 is divided by 5.
61 · 5 · Modular arithmetic & remai…
5. Find the remainder when 52 is divided by 8.
52 · 8 · Modular arithmetic & remai…
6. Find the remainder when 159 is divided by 12.
159 · 12 · Modular arithmetic & remai…
7. Find the remainder when 87 is divided by 5.
87 · 5 · Modular arithmetic & remai…
8. Find the remainder when 41 is divided by 6.
41 · 6 · Modular arithmetic & remai…
9. Find the remainder when 191 is divided by 12.
191 · 12 · Modular arithmetic & remai…
10. Find the remainder when 63 is divided by 5.
63 · 5 · Modular arithmetic & remai…
11. Find the remainder when 158 is divided by 11.
158 · 11 · Modular arithmetic & remai…
12. Find the remainder when 120 is divided by 11.
120 · 11 · Modular arithmetic & remai…

12 questions · Seed 104736 · Age guidance, not curriculum certification

Worksheet StudioAnswer key · Ages 16-18

Answers and reasoning. Review before teaching.

1. Find the remainder when 54 is divided by 5.
4
54 = 5 × 10 + 4, so the remainder is 4.
2. Find the remainder when 74 is divided by 5.
4
74 = 5 × 14 + 4, so the remainder is 4.
3. Find the remainder when 38 is divided by 5.
3
38 = 5 × 7 + 3, so the remainder is 3.
4. Find the remainder when 61 is divided by 5.
1
61 = 5 × 12 + 1, so the remainder is 1.
5. Find the remainder when 52 is divided by 8.
4
52 = 8 × 6 + 4, so the remainder is 4.
6. Find the remainder when 159 is divided by 12.
3
159 = 12 × 13 + 3, so the remainder is 3.
7. Find the remainder when 87 is divided by 5.
2
87 = 5 × 17 + 2, so the remainder is 2.
8. Find the remainder when 41 is divided by 6.
5
41 = 6 × 6 + 5, so the remainder is 5.
9. Find the remainder when 191 is divided by 12.
11
191 = 12 × 15 + 11, so the remainder is 11.
10. Find the remainder when 63 is divided by 5.
3
63 = 5 × 12 + 3, so the remainder is 3.
11. Find the remainder when 158 is divided by 11.
4
158 = 11 × 14 + 4, so the remainder is 4.
12. Find the remainder when 120 is divided by 11.
10
120 = 11 × 10 + 10, so the remainder is 10.

12 questions · Seed 104736 · Age guidance, not curriculum certification