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 45 is divided by 8.
45 · 8 · Modular arithmetic & remai…
2. Find the remainder when 215 is divided by 12.
215 · 12 · Modular arithmetic & remai…
3. Find the remainder when 32 is divided by 5.
32 · 5 · Modular arithmetic & remai…
4. Find the remainder when 90 is divided by 7.
90 · 7 · Modular arithmetic & remai…
5. Find the remainder when 102 is divided by 11.
102 · 11 · Modular arithmetic & remai…
6. Find the remainder when 83 is divided by 6.
83 · 6 · Modular arithmetic & remai…
7. Find the remainder when 91 is divided by 8.
91 · 8 · Modular arithmetic & remai…
8. Find the remainder when 59 is divided by 7.
59 · 7 · Modular arithmetic & remai…
9. Find the remainder when 77 is divided by 5.
77 · 5 · Modular arithmetic & remai…
10. Find the remainder when 130 is divided by 7.
130 · 7 · Modular arithmetic & remai…
11. Find the remainder when 39 is divided by 11.
39 · 11 · Modular arithmetic & remai…
12. Find the remainder when 114 is divided by 8.
114 · 8 · Modular arithmetic & remai…

12 questions · Seed 45 · 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 45 is divided by 8.
5
45 = 8 × 5 + 5, so the remainder is 5.
2. Find the remainder when 215 is divided by 12.
11
215 = 12 × 17 + 11, so the remainder is 11.
3. Find the remainder when 32 is divided by 5.
2
32 = 5 × 6 + 2, so the remainder is 2.
4. Find the remainder when 90 is divided by 7.
6
90 = 7 × 12 + 6, so the remainder is 6.
5. Find the remainder when 102 is divided by 11.
3
102 = 11 × 9 + 3, so the remainder is 3.
6. Find the remainder when 83 is divided by 6.
5
83 = 6 × 13 + 5, so the remainder is 5.
7. Find the remainder when 91 is divided by 8.
3
91 = 8 × 11 + 3, so the remainder is 3.
8. Find the remainder when 59 is divided by 7.
3
59 = 7 × 8 + 3, so the remainder is 3.
9. Find the remainder when 77 is divided by 5.
2
77 = 5 × 15 + 2, so the remainder is 2.
10. Find the remainder when 130 is divided by 7.
4
130 = 7 × 18 + 4, so the remainder is 4.
11. Find the remainder when 39 is divided by 11.
6
39 = 11 × 3 + 6, so the remainder is 6.
12. Find the remainder when 114 is divided by 8.
2
114 = 8 × 14 + 2, so the remainder is 2.

12 questions · Seed 45 · Age guidance, not curriculum certification

Worksheet StudioPractice pack · Ages 16-18

Take your time. Show your thinking.

1. Find the remainder when 45 is divided by 8.
45 · 8 · Modular arithmetic & remai…
2. Find the remainder when 215 is divided by 12.
215 · 12 · Modular arithmetic & remai…
3. Find the remainder when 32 is divided by 5.
32 · 5 · Modular arithmetic & remai…
4. Find the remainder when 90 is divided by 7.
90 · 7 · Modular arithmetic & remai…
5. Find the remainder when 102 is divided by 11.
102 · 11 · Modular arithmetic & remai…
6. Find the remainder when 83 is divided by 6.
83 · 6 · Modular arithmetic & remai…
7. Find the remainder when 91 is divided by 8.
91 · 8 · Modular arithmetic & remai…
8. Find the remainder when 59 is divided by 7.
59 · 7 · Modular arithmetic & remai…
9. Find the remainder when 77 is divided by 5.
77 · 5 · Modular arithmetic & remai…
10. Find the remainder when 130 is divided by 7.
130 · 7 · Modular arithmetic & remai…
11. Find the remainder when 39 is divided by 11.
39 · 11 · Modular arithmetic & remai…
12. Find the remainder when 114 is divided by 8.
114 · 8 · Modular arithmetic & remai…

12 questions · Seed 45 · Age guidance, not curriculum certification

Worksheet StudioAnswer key · Ages 16-18

Answers and reasoning. Review before teaching.

1. Find the remainder when 45 is divided by 8.
5
45 = 8 × 5 + 5, so the remainder is 5.
2. Find the remainder when 215 is divided by 12.
11
215 = 12 × 17 + 11, so the remainder is 11.
3. Find the remainder when 32 is divided by 5.
2
32 = 5 × 6 + 2, so the remainder is 2.
4. Find the remainder when 90 is divided by 7.
6
90 = 7 × 12 + 6, so the remainder is 6.
5. Find the remainder when 102 is divided by 11.
3
102 = 11 × 9 + 3, so the remainder is 3.
6. Find the remainder when 83 is divided by 6.
5
83 = 6 × 13 + 5, so the remainder is 5.
7. Find the remainder when 91 is divided by 8.
3
91 = 8 × 11 + 3, so the remainder is 3.
8. Find the remainder when 59 is divided by 7.
3
59 = 7 × 8 + 3, so the remainder is 3.
9. Find the remainder when 77 is divided by 5.
2
77 = 5 × 15 + 2, so the remainder is 2.
10. Find the remainder when 130 is divided by 7.
4
130 = 7 × 18 + 4, so the remainder is 4.
11. Find the remainder when 39 is divided by 11.
6
39 = 11 × 3 + 6, so the remainder is 6.
12. Find the remainder when 114 is divided by 8.
2
114 = 8 × 14 + 2, so the remainder is 2.

12 questions · Seed 45 · Age guidance, not curriculum certification