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 24 is divided by 5.
24 · 5 · Modular arithmetic & remai…
2. Find the remainder when 18 is divided by 5.
18 · 5 · Modular arithmetic & remai…
3. Find the remainder when 53 is divided by 12.
53 · 12 · Modular arithmetic & remai…
4. Find the remainder when 110 is divided by 9.
110 · 9 · Modular arithmetic & remai…
5. Find the remainder when 128 is divided by 7.
128 · 7 · Modular arithmetic & remai…
6. Find the remainder when 147 is divided by 11.
147 · 11 · Modular arithmetic & remai…
7. Find the remainder when 93 is divided by 7.
93 · 7 · Modular arithmetic & remai…
8. Find the remainder when 107 is divided by 12.
107 · 12 · Modular arithmetic & remai…
9. Find the remainder when 223 is divided by 12.
223 · 12 · Modular arithmetic & remai…
10. Find the remainder when 30 is divided by 7.
30 · 7 · Modular arithmetic & remai…
11. Find the remainder when 86 is divided by 7.
86 · 7 · Modular arithmetic & remai…
12. Find the remainder when 93 is divided by 9.
93 · 9 · Modular arithmetic & remai…

12 questions · Seed 46 · 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 24 is divided by 5.
4
24 = 5 × 4 + 4, so the remainder is 4.
2. Find the remainder when 18 is divided by 5.
3
18 = 5 × 3 + 3, so the remainder is 3.
3. Find the remainder when 53 is divided by 12.
5
53 = 12 × 4 + 5, so the remainder is 5.
4. Find the remainder when 110 is divided by 9.
2
110 = 9 × 12 + 2, so the remainder is 2.
5. Find the remainder when 128 is divided by 7.
2
128 = 7 × 18 + 2, so the remainder is 2.
6. Find the remainder when 147 is divided by 11.
4
147 = 11 × 13 + 4, so the remainder is 4.
7. Find the remainder when 93 is divided by 7.
2
93 = 7 × 13 + 2, so the remainder is 2.
8. Find the remainder when 107 is divided by 12.
11
107 = 12 × 8 + 11, so the remainder is 11.
9. Find the remainder when 223 is divided by 12.
7
223 = 12 × 18 + 7, so the remainder is 7.
10. Find the remainder when 30 is divided by 7.
2
30 = 7 × 4 + 2, so the remainder is 2.
11. Find the remainder when 86 is divided by 7.
2
86 = 7 × 12 + 2, so the remainder is 2.
12. Find the remainder when 93 is divided by 9.
3
93 = 9 × 10 + 3, so the remainder is 3.

12 questions · Seed 46 · Age guidance, not curriculum certification

Worksheet StudioPractice pack · Ages 13-15

Take your time. Show your thinking.

1. Find the remainder when 24 is divided by 5.
24 · 5 · Modular arithmetic & remai…
2. Find the remainder when 18 is divided by 5.
18 · 5 · Modular arithmetic & remai…
3. Find the remainder when 53 is divided by 12.
53 · 12 · Modular arithmetic & remai…
4. Find the remainder when 110 is divided by 9.
110 · 9 · Modular arithmetic & remai…
5. Find the remainder when 128 is divided by 7.
128 · 7 · Modular arithmetic & remai…
6. Find the remainder when 147 is divided by 11.
147 · 11 · Modular arithmetic & remai…
7. Find the remainder when 93 is divided by 7.
93 · 7 · Modular arithmetic & remai…
8. Find the remainder when 107 is divided by 12.
107 · 12 · Modular arithmetic & remai…
9. Find the remainder when 223 is divided by 12.
223 · 12 · Modular arithmetic & remai…
10. Find the remainder when 30 is divided by 7.
30 · 7 · Modular arithmetic & remai…
11. Find the remainder when 86 is divided by 7.
86 · 7 · Modular arithmetic & remai…
12. Find the remainder when 93 is divided by 9.
93 · 9 · Modular arithmetic & remai…

12 questions · Seed 46 · Age guidance, not curriculum certification

Worksheet StudioAnswer key · Ages 13-15

Answers and reasoning. Review before teaching.

1. Find the remainder when 24 is divided by 5.
4
24 = 5 × 4 + 4, so the remainder is 4.
2. Find the remainder when 18 is divided by 5.
3
18 = 5 × 3 + 3, so the remainder is 3.
3. Find the remainder when 53 is divided by 12.
5
53 = 12 × 4 + 5, so the remainder is 5.
4. Find the remainder when 110 is divided by 9.
2
110 = 9 × 12 + 2, so the remainder is 2.
5. Find the remainder when 128 is divided by 7.
2
128 = 7 × 18 + 2, so the remainder is 2.
6. Find the remainder when 147 is divided by 11.
4
147 = 11 × 13 + 4, so the remainder is 4.
7. Find the remainder when 93 is divided by 7.
2
93 = 7 × 13 + 2, so the remainder is 2.
8. Find the remainder when 107 is divided by 12.
11
107 = 12 × 8 + 11, so the remainder is 11.
9. Find the remainder when 223 is divided by 12.
7
223 = 12 × 18 + 7, so the remainder is 7.
10. Find the remainder when 30 is divided by 7.
2
30 = 7 × 4 + 2, so the remainder is 2.
11. Find the remainder when 86 is divided by 7.
2
86 = 7 × 12 + 2, so the remainder is 2.
12. Find the remainder when 93 is divided by 9.
3
93 = 9 × 10 + 3, so the remainder is 3.

12 questions · Seed 46 · Age guidance, not curriculum certification