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 155 is divided by 12.
155 · 12 · Modular arithmetic & remai…
2. Find the remainder when 31 is divided by 9.
31 · 9 · Modular arithmetic & remai…
3. Find the remainder when 97 is divided by 12.
97 · 12 · Modular arithmetic & remai…
4. Find the remainder when 125 is divided by 9.
125 · 9 · Modular arithmetic & remai…
5. Find the remainder when 104 is divided by 9.
104 · 9 · Modular arithmetic & remai…
6. Find the remainder when 68 is divided by 6.
68 · 6 · Modular arithmetic & remai…
7. Find the remainder when 199 is divided by 11.
199 · 11 · Modular arithmetic & remai…
8. Find the remainder when 142 is divided by 9.
142 · 9 · Modular arithmetic & remai…
9. Find the remainder when 146 is divided by 8.
146 · 8 · Modular arithmetic & remai…
10. Find the remainder when 28 is divided by 8.
28 · 8 · Modular arithmetic & remai…
11. Find the remainder when 67 is divided by 9.
67 · 9 · Modular arithmetic & remai…
12. Find the remainder when 94 is divided by 6.
94 · 6 · Modular arithmetic & remai…

12 questions · Seed 104732 · 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 155 is divided by 12.
11
155 = 12 × 12 + 11, so the remainder is 11.
2. Find the remainder when 31 is divided by 9.
4
31 = 9 × 3 + 4, so the remainder is 4.
3. Find the remainder when 97 is divided by 12.
1
97 = 12 × 8 + 1, so the remainder is 1.
4. Find the remainder when 125 is divided by 9.
8
125 = 9 × 13 + 8, so the remainder is 8.
5. Find the remainder when 104 is divided by 9.
5
104 = 9 × 11 + 5, so the remainder is 5.
6. Find the remainder when 68 is divided by 6.
2
68 = 6 × 11 + 2, so the remainder is 2.
7. Find the remainder when 199 is divided by 11.
1
199 = 11 × 18 + 1, so the remainder is 1.
8. Find the remainder when 142 is divided by 9.
7
142 = 9 × 15 + 7, so the remainder is 7.
9. Find the remainder when 146 is divided by 8.
2
146 = 8 × 18 + 2, so the remainder is 2.
10. Find the remainder when 28 is divided by 8.
4
28 = 8 × 3 + 4, so the remainder is 4.
11. Find the remainder when 67 is divided by 9.
4
67 = 9 × 7 + 4, so the remainder is 4.
12. Find the remainder when 94 is divided by 6.
4
94 = 6 × 15 + 4, so the remainder is 4.

12 questions · Seed 104732 · Age guidance, not curriculum certification

Worksheet StudioPractice pack · Ages 16-18

Take your time. Show your thinking.

1. Find the remainder when 155 is divided by 12.
155 · 12 · Modular arithmetic & remai…
2. Find the remainder when 31 is divided by 9.
31 · 9 · Modular arithmetic & remai…
3. Find the remainder when 97 is divided by 12.
97 · 12 · Modular arithmetic & remai…
4. Find the remainder when 125 is divided by 9.
125 · 9 · Modular arithmetic & remai…
5. Find the remainder when 104 is divided by 9.
104 · 9 · Modular arithmetic & remai…
6. Find the remainder when 68 is divided by 6.
68 · 6 · Modular arithmetic & remai…
7. Find the remainder when 199 is divided by 11.
199 · 11 · Modular arithmetic & remai…
8. Find the remainder when 142 is divided by 9.
142 · 9 · Modular arithmetic & remai…
9. Find the remainder when 146 is divided by 8.
146 · 8 · Modular arithmetic & remai…
10. Find the remainder when 28 is divided by 8.
28 · 8 · Modular arithmetic & remai…
11. Find the remainder when 67 is divided by 9.
67 · 9 · Modular arithmetic & remai…
12. Find the remainder when 94 is divided by 6.
94 · 6 · Modular arithmetic & remai…

12 questions · Seed 104732 · Age guidance, not curriculum certification

Worksheet StudioAnswer key · Ages 16-18

Answers and reasoning. Review before teaching.

1. Find the remainder when 155 is divided by 12.
11
155 = 12 × 12 + 11, so the remainder is 11.
2. Find the remainder when 31 is divided by 9.
4
31 = 9 × 3 + 4, so the remainder is 4.
3. Find the remainder when 97 is divided by 12.
1
97 = 12 × 8 + 1, so the remainder is 1.
4. Find the remainder when 125 is divided by 9.
8
125 = 9 × 13 + 8, so the remainder is 8.
5. Find the remainder when 104 is divided by 9.
5
104 = 9 × 11 + 5, so the remainder is 5.
6. Find the remainder when 68 is divided by 6.
2
68 = 6 × 11 + 2, so the remainder is 2.
7. Find the remainder when 199 is divided by 11.
1
199 = 11 × 18 + 1, so the remainder is 1.
8. Find the remainder when 142 is divided by 9.
7
142 = 9 × 15 + 7, so the remainder is 7.
9. Find the remainder when 146 is divided by 8.
2
146 = 8 × 18 + 2, so the remainder is 2.
10. Find the remainder when 28 is divided by 8.
4
28 = 8 × 3 + 4, so the remainder is 4.
11. Find the remainder when 67 is divided by 9.
4
67 = 9 × 7 + 4, so the remainder is 4.
12. Find the remainder when 94 is divided by 6.
4
94 = 6 × 15 + 4, so the remainder is 4.

12 questions · Seed 104732 · Age guidance, not curriculum certification