MATHCOUNTS Chapter Lab

Lesson 12: Modular Arithmetic & Cycles

Solve • Strategize • Check • Explain

Chapter-level practice with 50 multiple-choice problems and complete solutions.

MATHCOUNTS Chapter Lab — Lesson 12

Lesson Method

Use a contest-style checklist before choosing an answer.

1. Decode

Read for hidden constraints: integer, positive, inclusive, least, greatest.

2. Choose a Tool

Factorization, ratios, equations, diagrams, counting, or modular arithmetic.

3. Estimate

Use mental checks to eliminate unreasonable answers.

4. Solve Cleanly

Write only enough work to avoid mistakes under time pressure.

5. Verify

Check units, signs, parity, and whether the answer satisfies the question.

MATHCOUNTS Chapter Lab — Lesson 12

Topic Focus

The goal is fast recognition of structure, not long computation.

• Identify the mathematical structure before calculating.

• Use multiple approaches: arithmetic, algebra, diagrams and estimation.

• Keep choices hidden from your reasoning until you have a target answer.

• When stuck, test answer choices or use a smaller case.

MATHCOUNTS Chapter Lab — Lesson 12

Sprint Practice 1-2

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1. What is the remainder when 7 × 17 is divided by 10?��A. 0�B. 7�C. 9�D. 1

2. What is the sum of the digits of 7302?��A. 11�B. 12�C. 13�D. 24

MATHCOUNTS Chapter Lab — Lesson 12

Sprint Practice 3-4

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3. What is 5^5 × 5^5?��A. 298023223876953125�B. 6250�C. 9765625�D. 1953125

4. The sequence begins 6, 8, 10, ... . What is the 18th term?��A. 40�B. 20�C. 38�D. 42

MATHCOUNTS Chapter Lab — Lesson 12

Sprint Practice 5-6

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5. What is the least positive integer divisible by 16, 15 and 20?��A. 240�B. 120�C. 480�D. 256

6. What is the greatest common divisor of 72, 144 and 180?��A. 72�B. 42�C. 18�D. 36

MATHCOUNTS Chapter Lab — Lesson 12

Sprint Practice 7-8

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7. How many positive divisors does 600 have?��A. 24�B. 28�C. 48�D. 20

8. If 5x + 9 = 44, what is x?��A. 6�B. 7�C. 8�D. 9

MATHCOUNTS Chapter Lab — Lesson 12

Sprint Practice 9-10

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9. A bag has 6 red marbles and 5 blue marbles. What is the probability of drawing a red marble?��A. 6/5�B. 5/11�C. 6/12�D. 6/11

10. A lunch has 5 sandwich choices, 8 drink choices and 2 dessert choices. How many lunches are possible?��A. 15�B. 82�C. 80�D. 40

MATHCOUNTS Chapter Lab — Lesson 12

Sprint Practice 11-12

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11. What is the remainder when 8 × 23 is divided by 12?��A. 6�B. 5�C. 8�D. 4

12. What is the sum of the digits of 8672?��A. 24�B. 23�C. 22�D. 46

MATHCOUNTS Chapter Lab — Lesson 12

Sprint Practice 13-14

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13. What is 3^3 × 3^3?��A. 54�B. 243�C. 729�D. 19683

14. The sequence begins 6, 12, 18, ... . What is the 18th term?��A. 102�B. 114�C. 108�D. 24

MATHCOUNTS Chapter Lab — Lesson 12

Sprint Practice 15-16

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15. What is the least positive integer divisible by 16, 21 and 22?��A. 1848�B. 3712�C. 7392�D. 3696

16. What is the greatest common divisor of 72, 96 and 120?��A. 30�B. 24�C. 12�D. 48

MATHCOUNTS Chapter Lab — Lesson 12

Sprint Practice 17-18

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17. How many positive divisors does 1800 have?��A. 36�B. 32�C. 40�D. 72

18. If 5x + 5 = 55, what is x?��A. 11�B. 9�C. 10�D. 12

MATHCOUNTS Chapter Lab — Lesson 12

Sprint Practice 19-20

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19. A bag has 6 red marbles and 5 blue marbles. What is the probability of drawing a red marble?��A. 5/11�B. 6/5�C. 6/12�D. 6/11

20. A lunch has 3 sandwich choices, 8 drink choices and 3 dessert choices. How many lunches are possible?��A. 14�B. 72�C. 75�D. 24

MATHCOUNTS Chapter Lab — Lesson 12

Sprint Practice 21-22

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21. What is the remainder when 9 × 15 is divided by 6?��A. 4�B. 5�C. 3�D. 6

22. What is the sum of the digits of 10042?��A. 7�B. 14�C. 6�D. 8

MATHCOUNTS Chapter Lab — Lesson 12

Sprint Practice 23-24

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23. What is 5^4 × 5^5?��A. 390625�B. 3750�C. 1953125�D. 95367431640625

24. The sequence begins 6, 10, 14, ... . What is the 18th term?��A. 74�B. 78�C. 70�D. 22

MATHCOUNTS Chapter Lab — Lesson 12

Sprint Practice 25-26

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25. What is the least positive integer divisible by 16, 15 and 18?��A. 360�B. 736�C. 1440�D. 720

26. What is the greatest common divisor of 72, 144 and 150?��A. 3�B. 18�C. 12�D. 6

MATHCOUNTS Chapter Lab — Lesson 12

Sprint Practice 27-28

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27. How many positive divisors does 5400 have?��A. 52�B. 44�C. 96�D. 48

28. If 5x + 1 = 31, what is x?��A. 6�B. 7�C. 8�D. 5

MATHCOUNTS Chapter Lab — Lesson 12

Sprint Practice 29-30

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29. A bag has 6 red marbles and 5 blue marbles. What is the probability of drawing a red marble?��A. 6/5�B. 6/11�C. 6/12�D. 5/11

30. A lunch has 5 sandwich choices, 8 drink choices and 4 dessert choices. How many lunches are possible?��A. 160�B. 164�C. 17�D. 40

MATHCOUNTS Chapter Lab — Lesson 12

Sprint Practice 31-32

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31. What is the remainder when 10 × 21 is divided by 8?��A. 2�B. 3�C. 4�D. 6

32. What is the sum of the digits of 11412?��A. 18�B. 10�C. 8�D. 9

MATHCOUNTS Chapter Lab — Lesson 12

Sprint Practice 33-34

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33. What is 3^5 × 3^3?��A. 6561�B. 2187�C. 14348907�D. 270

34. The sequence begins 6, 8, 10, ... . What is the 18th term?��A. 38�B. 20�C. 42�D. 40

MATHCOUNTS Chapter Lab — Lesson 12

Sprint Practice 35-36

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35. What is the least positive integer divisible by 16, 21 and 20?��A. 840�B. 1680�C. 1696�D. 3360

36. What is the greatest common divisor of 72, 96 and 180?��A. 24�B. 18�C. 12�D. 6

MATHCOUNTS Chapter Lab — Lesson 12

Sprint Practice 37-38

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37. How many positive divisors does 600 have?��A. 48�B. 24�C. 20�D. 28

38. If 5x + 9 = 54, what is x?��A. 11�B. 9�C. 10�D. 8

MATHCOUNTS Chapter Lab — Lesson 12

Sprint Practice 39-40

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39. A bag has 6 red marbles and 5 blue marbles. What is the probability of drawing a red marble?��A. 6/5�B. 6/12�C. 5/11�D. 6/11

40. A lunch has 3 sandwich choices, 8 drink choices and 2 dessert choices. How many lunches are possible?��A. 50�B. 24�C. 13�D. 48

MATHCOUNTS Chapter Lab — Lesson 12

Sprint Practice 41-42

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41. What is the remainder when 11 × 13 is divided by 10?��A. 3�B. 4�C. 5�D. 7

42. What is the sum of the digits of 12782?��A. 40�B. 21�C. 19�D. 20

MATHCOUNTS Chapter Lab — Lesson 12

Sprint Practice 43-44

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43. What is 5^3 × 5^5?��A. 30517578125�B. 390625�C. 78125�D. 3250

44. The sequence begins 6, 12, 18, ... . What is the 18th term?��A. 102�B. 114�C. 108�D. 24

MATHCOUNTS Chapter Lab — Lesson 12

Sprint Practice 45-46

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45. What is the least positive integer divisible by 16, 15 and 22?��A. 1320�B. 2640�C. 2656�D. 5280

46. What is the greatest common divisor of 72, 144 and 120?��A. 24�B. 48�C. 30�D. 12

MATHCOUNTS Chapter Lab — Lesson 12

Sprint Practice 47-48

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47. How many positive divisors does 1800 have?��A. 72�B. 40�C. 36�D. 32

48. If 5x + 5 = 30, what is x?��A. 4�B. 5�C. 6�D. 7

MATHCOUNTS Chapter Lab — Lesson 12

Sprint Practice 49-50

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49. A bag has 6 red marbles and 5 blue marbles. What is the probability of drawing a red marble?��A. 6/11�B. 6/12�C. 5/11�D. 6/5

50. A lunch has 5 sandwich choices, 8 drink choices and 3 dessert choices. How many lunches are possible?��A. 40�B. 123�C. 16�D. 120

MATHCOUNTS Chapter Lab — Lesson 12

Answer Key

Multiple-choice questions 1-50

1. C�2. B�3. C�4. A�5. A�6. D�7. A�8. B�9. D�10. C

11. D�12. B�13. C�14. C�15. D�16. B�17. A�18. C�19. D�20. B

21. C�22. A�23. C�24. A�25. D�26. D�27. D�28. A�29. B�30. A

31. A�32. D�33. A�34. D�35. B�36. C�37. B�38. B�39. D�40. D

41. A�42. D�43. B�44. C�45. B�46. A�47. C�48. B�49. A�50. D

MATHCOUNTS Chapter Lab — Lesson 12

Solutions 1-2

Read after completing the practice section.

1. Answer: C. 9�Reduce modulo 10: 7 ≡ 7 and 17 ≡ 7. The product remainder is 49 mod 10 = 9.

2. Answer: B. 12�Add the digits of 7302: 7 + 3 + 0 + 2 = 12.

MATHCOUNTS Chapter Lab — Lesson 12

Solutions 3-4

Read after completing the practice section.

3. Answer: C. 9765625�When multiplying powers with the same base, add exponents: 5^5×5^5=5^10=9765625.

4. Answer: A. 40�This is arithmetic with first term 6 and common difference 2. Term 18 is 6+(18-1)2 = 40.

MATHCOUNTS Chapter Lab — Lesson 12

Solutions 5-6

Read after completing the practice section.

5. Answer: A. 240�The least positive integer divisible by all three numbers is their LCM. Using prime factors gives LCM = 240.

6. Answer: D. 36�Find the greatest number dividing all three. Repeated gcd gives gcd(72,144,180) = 36.

MATHCOUNTS Chapter Lab — Lesson 12

Solutions 7-8

Read after completing the practice section.

7. Answer: A. 24�Factor 600 = 2^3 · 3 · 5^2. The number of positive divisors is the product of one more than each exponent: 4 × 2 × 3 = 24.

8. Answer: B. 7�Subtract 9: 5x = 35. Divide by 5: x = 7.

MATHCOUNTS Chapter Lab — Lesson 12

Solutions 9-10

Read after completing the practice section.

9. Answer: D. 6/11�There are 11 total marbles and 6 favorable red marbles. Probability = 6/11.

10. Answer: C. 80�Use the multiplication principle: 5 × 8 × 2 = 80 possible lunches.

MATHCOUNTS Chapter Lab — Lesson 12

Solutions 11-12

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11. Answer: D. 4�Reduce modulo 12: 8 ≡ 8 and 23 ≡ 11. The product remainder is 88 mod 12 = 4.

12. Answer: B. 23�Add the digits of 8672: 8 + 6 + 7 + 2 = 23.

MATHCOUNTS Chapter Lab — Lesson 12

Solutions 13-14

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13. Answer: C. 729�When multiplying powers with the same base, add exponents: 3^3×3^3=3^6=729.

14. Answer: C. 108�This is arithmetic with first term 6 and common difference 6. Term 18 is 6+(18-1)6 = 108.

MATHCOUNTS Chapter Lab — Lesson 12

Solutions 15-16

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15. Answer: D. 3696�The least positive integer divisible by all three numbers is their LCM. Using prime factors gives LCM = 3696.

16. Answer: B. 24�Find the greatest number dividing all three. Repeated gcd gives gcd(72,96,120) = 24.

MATHCOUNTS Chapter Lab — Lesson 12

Solutions 17-18

Read after completing the practice section.

17. Answer: A. 36�Factor 1800 = 2^3 · 3^2 · 5^2. The number of positive divisors is the product of one more than each exponent: 4 × 3 × 3 = 36.

18. Answer: C. 10�Subtract 5: 5x = 50. Divide by 5: x = 10.

MATHCOUNTS Chapter Lab — Lesson 12

Solutions 19-20

Read after completing the practice section.

19. Answer: D. 6/11�There are 11 total marbles and 6 favorable red marbles. Probability = 6/11.

20. Answer: B. 72�Use the multiplication principle: 3 × 8 × 3 = 72 possible lunches.

MATHCOUNTS Chapter Lab — Lesson 12

Solutions 21-22

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21. Answer: C. 3�Reduce modulo 6: 9 ≡ 3 and 15 ≡ 3. The product remainder is 9 mod 6 = 3.

22. Answer: A. 7�Add the digits of 10042: 1 + 0 + 0 + 4 + 2 = 7.

MATHCOUNTS Chapter Lab — Lesson 12

Solutions 23-24

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23. Answer: C. 1953125�When multiplying powers with the same base, add exponents: 5^4×5^5=5^9=1953125.

24. Answer: A. 74�This is arithmetic with first term 6 and common difference 4. Term 18 is 6+(18-1)4 = 74.

MATHCOUNTS Chapter Lab — Lesson 12

Solutions 25-26

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25. Answer: D. 720�The least positive integer divisible by all three numbers is their LCM. Using prime factors gives LCM = 720.

26. Answer: D. 6�Find the greatest number dividing all three. Repeated gcd gives gcd(72,144,150) = 6.

MATHCOUNTS Chapter Lab — Lesson 12

Solutions 27-28

Read after completing the practice section.

27. Answer: D. 48�Factor 5400 = 2^3 · 3^3 · 5^2. The number of positive divisors is the product of one more than each exponent: 4 × 4 × 3 = 48.

28. Answer: A. 6�Subtract 1: 5x = 30. Divide by 5: x = 6.

MATHCOUNTS Chapter Lab — Lesson 12

Solutions 29-30

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29. Answer: B. 6/11�There are 11 total marbles and 6 favorable red marbles. Probability = 6/11.

30. Answer: A. 160�Use the multiplication principle: 5 × 8 × 4 = 160 possible lunches.

MATHCOUNTS Chapter Lab — Lesson 12

Solutions 31-32

Read after completing the practice section.

31. Answer: A. 2�Reduce modulo 8: 10 ≡ 2 and 21 ≡ 5. The product remainder is 10 mod 8 = 2.

32. Answer: D. 9�Add the digits of 11412: 1 + 1 + 4 + 1 + 2 = 9.

MATHCOUNTS Chapter Lab — Lesson 12

Solutions 33-34

Read after completing the practice section.

33. Answer: A. 6561�When multiplying powers with the same base, add exponents: 3^5×3^3=3^8=6561.

34. Answer: D. 40�This is arithmetic with first term 6 and common difference 2. Term 18 is 6+(18-1)2 = 40.

MATHCOUNTS Chapter Lab — Lesson 12

Solutions 35-36

Read after completing the practice section.

35. Answer: B. 1680�The least positive integer divisible by all three numbers is their LCM. Using prime factors gives LCM = 1680.

36. Answer: C. 12�Find the greatest number dividing all three. Repeated gcd gives gcd(72,96,180) = 12.

MATHCOUNTS Chapter Lab — Lesson 12

Solutions 37-38

Read after completing the practice section.

37. Answer: B. 24�Factor 600 = 2^3 · 3 · 5^2. The number of positive divisors is the product of one more than each exponent: 4 × 2 × 3 = 24.

38. Answer: B. 9�Subtract 9: 5x = 45. Divide by 5: x = 9.

MATHCOUNTS Chapter Lab — Lesson 12

Solutions 39-40

Read after completing the practice section.

39. Answer: D. 6/11�There are 11 total marbles and 6 favorable red marbles. Probability = 6/11.

40. Answer: D. 48�Use the multiplication principle: 3 × 8 × 2 = 48 possible lunches.

MATHCOUNTS Chapter Lab — Lesson 12

Solutions 41-42

Read after completing the practice section.

41. Answer: A. 3�Reduce modulo 10: 11 ≡ 1 and 13 ≡ 3. The product remainder is 3 mod 10 = 3.

42. Answer: D. 20�Add the digits of 12782: 1 + 2 + 7 + 8 + 2 = 20.

MATHCOUNTS Chapter Lab — Lesson 12

Solutions 43-44

Read after completing the practice section.

43. Answer: B. 390625�When multiplying powers with the same base, add exponents: 5^3×5^5=5^8=390625.

44. Answer: C. 108�This is arithmetic with first term 6 and common difference 6. Term 18 is 6+(18-1)6 = 108.

MATHCOUNTS Chapter Lab — Lesson 12

Solutions 45-46

Read after completing the practice section.

45. Answer: B. 2640�The least positive integer divisible by all three numbers is their LCM. Using prime factors gives LCM = 2640.

46. Answer: A. 24�Find the greatest number dividing all three. Repeated gcd gives gcd(72,144,120) = 24.

MATHCOUNTS Chapter Lab — Lesson 12

Solutions 47-48

Read after completing the practice section.

47. Answer: C. 36�Factor 1800 = 2^3 · 3^2 · 5^2. The number of positive divisors is the product of one more than each exponent: 4 × 3 × 3 = 36.

48. Answer: B. 5�Subtract 5: 5x = 25. Divide by 5: x = 5.

MATHCOUNTS Chapter Lab — Lesson 12

Solutions 49-50

Read after completing the practice section.

49. Answer: A. 6/11�There are 11 total marbles and 6 favorable red marbles. Probability = 6/11.

50. Answer: D. 120�Use the multiplication principle: 5 × 8 × 3 = 120 possible lunches.

MATHCOUNTS Chapter Lab — Lesson 12