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 22 is divided by 6.
22 · 6 · Modular arithmetic & remai…
2. Find the remainder when 74 is divided by 7.
74 · 7 · Modular arithmetic & remai…
3. Find the remainder when 162 is divided by 12.
162 · 12 · Modular arithmetic & remai…
4. Find the remainder when 68 is divided by 6.
68 · 6 · Modular arithmetic & remai…
5. Find the remainder when 37 is divided by 5.
37 · 5 · Modular arithmetic & remai…
6. Find the remainder when 48 is divided by 5.
48 · 5 · Modular arithmetic & remai…
7. Find the remainder when 129 is divided by 11.
129 · 11 · Modular arithmetic & remai…
8. Find the remainder when 75 is divided by 9.
75 · 9 · Modular arithmetic & remai…
9. Find the remainder when 47 is divided by 8.
47 · 8 · Modular arithmetic & remai…
10. Find the remainder when 120 is divided by 7.
120 · 7 · Modular arithmetic & remai…
11. Find the remainder when 86 is divided by 5.
86 · 5 · Modular arithmetic & remai…
12. Find the remainder when 69 is divided by 7.
69 · 7 · Modular arithmetic & remai…

12 questions · Seed 104734 · 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 22 is divided by 6.
4
22 = 6 × 3 + 4, so the remainder is 4.
2. Find the remainder when 74 is divided by 7.
4
74 = 7 × 10 + 4, so the remainder is 4.
3. Find the remainder when 162 is divided by 12.
6
162 = 12 × 13 + 6, so the remainder is 6.
4. Find the remainder when 68 is divided by 6.
2
68 = 6 × 11 + 2, so the remainder is 2.
5. Find the remainder when 37 is divided by 5.
2
37 = 5 × 7 + 2, so the remainder is 2.
6. Find the remainder when 48 is divided by 5.
3
48 = 5 × 9 + 3, so the remainder is 3.
7. Find the remainder when 129 is divided by 11.
8
129 = 11 × 11 + 8, so the remainder is 8.
8. Find the remainder when 75 is divided by 9.
3
75 = 9 × 8 + 3, so the remainder is 3.
9. Find the remainder when 47 is divided by 8.
7
47 = 8 × 5 + 7, so the remainder is 7.
10. Find the remainder when 120 is divided by 7.
1
120 = 7 × 17 + 1, so the remainder is 1.
11. Find the remainder when 86 is divided by 5.
1
86 = 5 × 17 + 1, so the remainder is 1.
12. Find the remainder when 69 is divided by 7.
6
69 = 7 × 9 + 6, so the remainder is 6.

12 questions · Seed 104734 · Age guidance, not curriculum certification

Worksheet StudioPractice pack · Ages 16-18

Take your time. Show your thinking.

1. Find the remainder when 22 is divided by 6.
22 · 6 · Modular arithmetic & remai…
2. Find the remainder when 74 is divided by 7.
74 · 7 · Modular arithmetic & remai…
3. Find the remainder when 162 is divided by 12.
162 · 12 · Modular arithmetic & remai…
4. Find the remainder when 68 is divided by 6.
68 · 6 · Modular arithmetic & remai…
5. Find the remainder when 37 is divided by 5.
37 · 5 · Modular arithmetic & remai…
6. Find the remainder when 48 is divided by 5.
48 · 5 · Modular arithmetic & remai…
7. Find the remainder when 129 is divided by 11.
129 · 11 · Modular arithmetic & remai…
8. Find the remainder when 75 is divided by 9.
75 · 9 · Modular arithmetic & remai…
9. Find the remainder when 47 is divided by 8.
47 · 8 · Modular arithmetic & remai…
10. Find the remainder when 120 is divided by 7.
120 · 7 · Modular arithmetic & remai…
11. Find the remainder when 86 is divided by 5.
86 · 5 · Modular arithmetic & remai…
12. Find the remainder when 69 is divided by 7.
69 · 7 · Modular arithmetic & remai…

12 questions · Seed 104734 · Age guidance, not curriculum certification

Worksheet StudioAnswer key · Ages 16-18

Answers and reasoning. Review before teaching.

1. Find the remainder when 22 is divided by 6.
4
22 = 6 × 3 + 4, so the remainder is 4.
2. Find the remainder when 74 is divided by 7.
4
74 = 7 × 10 + 4, so the remainder is 4.
3. Find the remainder when 162 is divided by 12.
6
162 = 12 × 13 + 6, so the remainder is 6.
4. Find the remainder when 68 is divided by 6.
2
68 = 6 × 11 + 2, so the remainder is 2.
5. Find the remainder when 37 is divided by 5.
2
37 = 5 × 7 + 2, so the remainder is 2.
6. Find the remainder when 48 is divided by 5.
3
48 = 5 × 9 + 3, so the remainder is 3.
7. Find the remainder when 129 is divided by 11.
8
129 = 11 × 11 + 8, so the remainder is 8.
8. Find the remainder when 75 is divided by 9.
3
75 = 9 × 8 + 3, so the remainder is 3.
9. Find the remainder when 47 is divided by 8.
7
47 = 8 × 5 + 7, so the remainder is 7.
10. Find the remainder when 120 is divided by 7.
1
120 = 7 × 17 + 1, so the remainder is 1.
11. Find the remainder when 86 is divided by 5.
1
86 = 5 × 17 + 1, so the remainder is 1.
12. Find the remainder when 69 is divided by 7.
6
69 = 7 × 9 + 6, so the remainder is 6.

12 questions · Seed 104734 · Age guidance, not curriculum certification