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 69 is divided by 9.
69 · 9 · Modular arithmetic & remai…
2. Find the remainder when 54 is divided by 8.
54 · 8 · Modular arithmetic & remai…
3. Find the remainder when 68 is divided by 5.
68 · 5 · Modular arithmetic & remai…
4. Find the remainder when 69 is divided by 11.
69 · 11 · Modular arithmetic & remai…
5. Find the remainder when 24 is divided by 5.
24 · 5 · Modular arithmetic & remai…
6. Find the remainder when 113 is divided by 12.
113 · 12 · Modular arithmetic & remai…
7. Find the remainder when 101 is divided by 9.
101 · 9 · Modular arithmetic & remai…
8. Find the remainder when 83 is divided by 6.
83 · 6 · Modular arithmetic & remai…
9. Find the remainder when 116 is divided by 9.
116 · 9 · Modular arithmetic & remai…
10. Find the remainder when 27 is divided by 8.
27 · 8 · Modular arithmetic & remai…
11. Find the remainder when 60 is divided by 8.
60 · 8 · Modular arithmetic & remai…
12. Find the remainder when 109 is divided by 6.
109 · 6 · Modular arithmetic & remai…

12 questions · Seed 47 · 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 69 is divided by 9.
6
69 = 9 × 7 + 6, so the remainder is 6.
2. Find the remainder when 54 is divided by 8.
6
54 = 8 × 6 + 6, so the remainder is 6.
3. Find the remainder when 68 is divided by 5.
3
68 = 5 × 13 + 3, so the remainder is 3.
4. Find the remainder when 69 is divided by 11.
3
69 = 11 × 6 + 3, so the remainder is 3.
5. Find the remainder when 24 is divided by 5.
4
24 = 5 × 4 + 4, so the remainder is 4.
6. Find the remainder when 113 is divided by 12.
5
113 = 12 × 9 + 5, so the remainder is 5.
7. Find the remainder when 101 is divided by 9.
2
101 = 9 × 11 + 2, so the remainder is 2.
8. Find the remainder when 83 is divided by 6.
5
83 = 6 × 13 + 5, so the remainder is 5.
9. Find the remainder when 116 is divided by 9.
8
116 = 9 × 12 + 8, so the remainder is 8.
10. Find the remainder when 27 is divided by 8.
3
27 = 8 × 3 + 3, so the remainder is 3.
11. Find the remainder when 60 is divided by 8.
4
60 = 8 × 7 + 4, so the remainder is 4.
12. Find the remainder when 109 is divided by 6.
1
109 = 6 × 18 + 1, so the remainder is 1.

12 questions · Seed 47 · Age guidance, not curriculum certification

Worksheet StudioPractice pack · Ages 13-15

Take your time. Show your thinking.

1. Find the remainder when 69 is divided by 9.
69 · 9 · Modular arithmetic & remai…
2. Find the remainder when 54 is divided by 8.
54 · 8 · Modular arithmetic & remai…
3. Find the remainder when 68 is divided by 5.
68 · 5 · Modular arithmetic & remai…
4. Find the remainder when 69 is divided by 11.
69 · 11 · Modular arithmetic & remai…
5. Find the remainder when 24 is divided by 5.
24 · 5 · Modular arithmetic & remai…
6. Find the remainder when 113 is divided by 12.
113 · 12 · Modular arithmetic & remai…
7. Find the remainder when 101 is divided by 9.
101 · 9 · Modular arithmetic & remai…
8. Find the remainder when 83 is divided by 6.
83 · 6 · Modular arithmetic & remai…
9. Find the remainder when 116 is divided by 9.
116 · 9 · Modular arithmetic & remai…
10. Find the remainder when 27 is divided by 8.
27 · 8 · Modular arithmetic & remai…
11. Find the remainder when 60 is divided by 8.
60 · 8 · Modular arithmetic & remai…
12. Find the remainder when 109 is divided by 6.
109 · 6 · Modular arithmetic & remai…

12 questions · Seed 47 · Age guidance, not curriculum certification

Worksheet StudioAnswer key · Ages 13-15

Answers and reasoning. Review before teaching.

1. Find the remainder when 69 is divided by 9.
6
69 = 9 × 7 + 6, so the remainder is 6.
2. Find the remainder when 54 is divided by 8.
6
54 = 8 × 6 + 6, so the remainder is 6.
3. Find the remainder when 68 is divided by 5.
3
68 = 5 × 13 + 3, so the remainder is 3.
4. Find the remainder when 69 is divided by 11.
3
69 = 11 × 6 + 3, so the remainder is 3.
5. Find the remainder when 24 is divided by 5.
4
24 = 5 × 4 + 4, so the remainder is 4.
6. Find the remainder when 113 is divided by 12.
5
113 = 12 × 9 + 5, so the remainder is 5.
7. Find the remainder when 101 is divided by 9.
2
101 = 9 × 11 + 2, so the remainder is 2.
8. Find the remainder when 83 is divided by 6.
5
83 = 6 × 13 + 5, so the remainder is 5.
9. Find the remainder when 116 is divided by 9.
8
116 = 9 × 12 + 8, so the remainder is 8.
10. Find the remainder when 27 is divided by 8.
3
27 = 8 × 3 + 3, so the remainder is 3.
11. Find the remainder when 60 is divided by 8.
4
60 = 8 × 7 + 4, so the remainder is 4.
12. Find the remainder when 109 is divided by 6.
1
109 = 6 × 18 + 1, so the remainder is 1.

12 questions · Seed 47 · Age guidance, not curriculum certification