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.

Choose “Save as PDF” to keep a copy. Use your browser’s paper setting for A4 or Letter.

Worksheet StudioPractice pack · Ages 13-15

Take your time. Show your thinking.

1. Find the remainder when 28 is divided by 6.
28 · 6 · Modular arithmetic & remai…
2. Find the remainder when 131 is divided by 11.
131 · 11 · Modular arithmetic & remai…
3. Find the remainder when 149 is divided by 8.
149 · 8 · Modular arithmetic & remai…
4. Find the remainder when 107 is divided by 8.
107 · 8 · Modular arithmetic & remai…
5. Find the remainder when 68 is divided by 5.
68 · 5 · Modular arithmetic & remai…
6. Find the remainder when 67 is divided by 7.
67 · 7 · Modular arithmetic & remai…
7. Find the remainder when 90 is divided by 7.
90 · 7 · Modular arithmetic & remai…
8. Find the remainder when 199 is divided by 12.
199 · 12 · Modular arithmetic & remai…
9. Find the remainder when 104 is divided by 11.
104 · 11 · Modular arithmetic & remai…
10. Find the remainder when 67 is divided by 8.
67 · 8 · Modular arithmetic & remai…
11. Find the remainder when 120 is divided by 9.
120 · 9 · Modular arithmetic & remai…
12. Find the remainder when 53 is divided by 6.
53 · 6 · Modular arithmetic & remai…

12 questions · Seed 104739 · 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 28 is divided by 6.
4
28 = 6 × 4 + 4, so the remainder is 4.
2. Find the remainder when 131 is divided by 11.
10
131 = 11 × 11 + 10, so the remainder is 10.
3. Find the remainder when 149 is divided by 8.
5
149 = 8 × 18 + 5, so the remainder is 5.
4. Find the remainder when 107 is divided by 8.
3
107 = 8 × 13 + 3, so the remainder is 3.
5. Find the remainder when 68 is divided by 5.
3
68 = 5 × 13 + 3, so the remainder is 3.
6. Find the remainder when 67 is divided by 7.
4
67 = 7 × 9 + 4, so the remainder is 4.
7. Find the remainder when 90 is divided by 7.
6
90 = 7 × 12 + 6, so the remainder is 6.
8. Find the remainder when 199 is divided by 12.
7
199 = 12 × 16 + 7, so the remainder is 7.
9. Find the remainder when 104 is divided by 11.
5
104 = 11 × 9 + 5, so the remainder is 5.
10. Find the remainder when 67 is divided by 8.
3
67 = 8 × 8 + 3, so the remainder is 3.
11. Find the remainder when 120 is divided by 9.
3
120 = 9 × 13 + 3, so the remainder is 3.
12. Find the remainder when 53 is divided by 6.
5
53 = 6 × 8 + 5, so the remainder is 5.

12 questions · Seed 104739 · Age guidance, not curriculum certification

Worksheet StudioPractice pack · Ages 13-15

Take your time. Show your thinking.

1. Find the remainder when 28 is divided by 6.
28 · 6 · Modular arithmetic & remai…
2. Find the remainder when 131 is divided by 11.
131 · 11 · Modular arithmetic & remai…
3. Find the remainder when 149 is divided by 8.
149 · 8 · Modular arithmetic & remai…
4. Find the remainder when 107 is divided by 8.
107 · 8 · Modular arithmetic & remai…
5. Find the remainder when 68 is divided by 5.
68 · 5 · Modular arithmetic & remai…
6. Find the remainder when 67 is divided by 7.
67 · 7 · Modular arithmetic & remai…
7. Find the remainder when 90 is divided by 7.
90 · 7 · Modular arithmetic & remai…
8. Find the remainder when 199 is divided by 12.
199 · 12 · Modular arithmetic & remai…
9. Find the remainder when 104 is divided by 11.
104 · 11 · Modular arithmetic & remai…
10. Find the remainder when 67 is divided by 8.
67 · 8 · Modular arithmetic & remai…
11. Find the remainder when 120 is divided by 9.
120 · 9 · Modular arithmetic & remai…
12. Find the remainder when 53 is divided by 6.
53 · 6 · Modular arithmetic & remai…

12 questions · Seed 104739 · Age guidance, not curriculum certification

Worksheet StudioAnswer key · Ages 13-15

Answers and reasoning. Review before teaching.

1. Find the remainder when 28 is divided by 6.
4
28 = 6 × 4 + 4, so the remainder is 4.
2. Find the remainder when 131 is divided by 11.
10
131 = 11 × 11 + 10, so the remainder is 10.
3. Find the remainder when 149 is divided by 8.
5
149 = 8 × 18 + 5, so the remainder is 5.
4. Find the remainder when 107 is divided by 8.
3
107 = 8 × 13 + 3, so the remainder is 3.
5. Find the remainder when 68 is divided by 5.
3
68 = 5 × 13 + 3, so the remainder is 3.
6. Find the remainder when 67 is divided by 7.
4
67 = 7 × 9 + 4, so the remainder is 4.
7. Find the remainder when 90 is divided by 7.
6
90 = 7 × 12 + 6, so the remainder is 6.
8. Find the remainder when 199 is divided by 12.
7
199 = 12 × 16 + 7, so the remainder is 7.
9. Find the remainder when 104 is divided by 11.
5
104 = 11 × 9 + 5, so the remainder is 5.
10. Find the remainder when 67 is divided by 8.
3
67 = 8 × 8 + 3, so the remainder is 3.
11. Find the remainder when 120 is divided by 9.
3
120 = 9 × 13 + 3, so the remainder is 3.
12. Find the remainder when 53 is divided by 6.
5
53 = 6 × 8 + 5, so the remainder is 5.

12 questions · Seed 104739 · Age guidance, not curriculum certification