Factors, Multiples & Primes

87 Class 6 IMO Mathematics Olympiad practice questions on factors, multiples & primes, part of the Number Sense topic area. Every question below includes the correct answer and a full explanation.

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1. Which of these is a factor of 24?
- A.5
- B.7
- C.8
- D.10
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Answer: C. 8
24 ÷ 8 = 3 exactly, so 8 is a factor of 24.
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2. Which of these numbers is a multiple of 6?
- A.16
- B.18
- C.20
- D.22
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Answer: B. 18
18 = 6 × 3.
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3. What is the smallest positive number that must be added to 754 to make it divisible by 9?
- A.4
- B.3
- C.1
- D.2
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Answer: D. 2
754 ÷ 9 leaves a remainder, and adding 2 reaches the next multiple of 9.
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4. What is the LCM of 13 and 10?
- A.143
- B.140
- C.126
- D.130
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Answer: D. 130
The smallest number that both 13 and 10 divide into exactly is 130.
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5. What is the HCF of 27 and 48?
- A.1
- B.4
- C.6
- D.3
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Answer: D. 3
The highest number that divides both 27 and 48 exactly is 3.
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6. What is the smallest positive number that must be added to 824 to make it divisible by 9?
- A.6
- B.3
- C.4
- D.5
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Answer: C. 4
824 ÷ 9 leaves a remainder, and adding 4 reaches the next multiple of 9.
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7. What is the remainder when 154 is divided by 7?
- A.3
- B.1
- C.2
- D.0
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Answer: D. 0
154 ÷ 7 leaves a remainder of 0.
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8. What is the remainder when 104 is divided by 10?
- A.6
- B.4
- C.7
- D.5
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Answer: B. 4
104 ÷ 10 leaves a remainder of 4.
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9. What is the LCM of 13 and 7?
- A.87
- B.98
- C.104
- D.91
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Answer: D. 91
The smallest number that both 13 and 7 divide into exactly is 91.
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10. What is the LCM of 6 and 8?
- A.30
- B.20
- C.32
- D.24
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Answer: D. 24
The smallest number that both 6 and 8 divide into exactly is 24.
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11. What is the HCF of 56 and 21?
- A.8
- B.7
- C.5
- D.14
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Answer: B. 7
The highest number that divides both 56 and 21 exactly is 7.
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12. What is the LCM of 10 and 6?
- A.26
- B.30
- C.40
- D.36
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Answer: B. 30
The smallest number that both 10 and 6 divide into exactly is 30.
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13. How many factors does 60 have?
- A.13
- B.14
- C.12
- D.11
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Answer: C. 12
60 has 12 factors in total.
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14. What is the HCF of 63 and 90?
- A.9
- B.18
- C.6
- D.12
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Answer: A. 9
The highest number that divides both 63 and 90 exactly is 9.
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15. What is the LCM of 30 and 12?
- A.60
- B.54
- C.66
- D.90
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Answer: A. 60
The smallest number that both 30 and 12 divide into exactly is 60.
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16. What is the LCM of 36 and 12?
- A.30
- B.36
- C.42
- D.72
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Answer: B. 36
The smallest number that both 36 and 12 divide into exactly is 36.
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17. What is the HCF of 75 and 90?
- A.12
- B.30
- C.15
- D.18
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Answer: C. 15
The highest number that divides both 75 and 90 exactly is 15.
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18. Which pair of numbers is a pair of twin primes (prime numbers differing by 2)?
- A.(3, 5)
- B.(5, 9)
- C.(7, 10)
- D.(11, 14)
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Answer: A. (3, 5)
3 and 5 are both prime and differ by 2. The other pairs each contain a non-prime number.
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19. Which of the following pairs of numbers is co-prime (shares no common factor other than 1)?
- A.(8, 9)
- B.(6, 9)
- C.(4, 8)
- D.(9, 12)
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Answer: A. (8, 9)
8 = 2³ and 9 = 3² share no common factor besides 1, so they're co-prime. The other pairs all share a factor greater than 1.
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20. What is the HCF of 8 and 12?
- A.2
- B.4
- C.6
- D.8
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Answer: B. 4
Factors of 8: 1,2,4,8. Factors of 12: 1,2,3,4,6,12. Highest common factor is 4.
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21. What is the LCM of 4 and 5?
- A.9
- B.10
- C.20
- D.40
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Answer: C. 20
Multiples of 4: 4,8,12,16,20. Multiples of 5: 5,10,15,20. Smallest common multiple is 20.
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22. Which number is prime?
- A.21
- B.27
- C.29
- D.33
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Answer: C. 29
21=3×7, 27=3×3×3, 33=3×11 — all have factors besides 1 and themselves. 29 has none.
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23. Which of the following is NOT a factor of 36?
- A.4
- B.6
- C.8
- D.9
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Answer: C. 8
36 ÷ 8 = 4.5, not a whole number.
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24. What is the HCF of 15 and 25?
- A.3
- B.5
- C.10
- D.15
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Answer: B. 5
Factors of 15: 1,3,5,15. Factors of 25: 1,5,25. Common factors: 1,5 — HCF is 5.
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25. The HCF of 36 and 48 is:
- A.6
- B.8
- C.12
- D.16
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Answer: C. 12
36 = 2²×3², 48 = 2⁴×3. HCF = 2²×3 = 12.
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26. The LCM of 8 and 12 is:
- A.16
- B.20
- C.24
- D.48
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Answer: C. 24
Multiples of 8: 8,16,24. Multiples of 12: 12,24. LCM = 24.
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27. Which number is prime?
- A.51
- B.57
- C.61
- D.69
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Answer: C. 61
51=3×17, 57=3×19, 69=3×23 — all composite. 61 has no factors besides 1 and itself.
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28. What is the remainder when 157 is divided by 12?
- A.1
- B.11
- C.12
- D.13
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Answer: A. 1
12 × 13 = 156, and 157 − 156 = 1.
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29. A number is divisible by 2, 3 and 5. Which is the smallest such number?
- A.10
- B.20
- C.30
- D.60
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Answer: C. 30
LCM(2,3,5) = 30.
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30. What is the remainder when 109 is divided by 4?
- A.3
- B.2
- C.1
- D.0
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Answer: C. 1
109 ÷ 4 leaves a remainder of 1.
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31. What is the remainder when 151 is divided by 12?
- A.8
- B.10
- C.7
- D.9
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Answer: C. 7
151 ÷ 12 leaves a remainder of 7.
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32. What is the LCM of 5 and 12?
- A.60
- B.56
- C.72
- D.65
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Answer: A. 60
The smallest number that both 5 and 12 divide into exactly is 60.
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33. What is the LCM of 7 and 12?
- A.96
- B.91
- C.84
- D.80
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Answer: C. 84
The smallest number that both 7 and 12 divide into exactly is 84.
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34. What is the remainder when 119 is divided by 8?
- A.2
- B.0
- C.7
- D.1
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Answer: C. 7
119 ÷ 8 leaves a remainder of 7.
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35. What is the remainder when 141 is divided by 7?
- A.4
- B.3
- C.2
- D.1
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Answer: D. 1
141 ÷ 7 leaves a remainder of 1.
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36. What is the HCF of 81 and 90?
- A.6
- B.12
- C.9
- D.18
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Answer: C. 9
The highest number that divides both 81 and 90 exactly is 9.
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37. What is the LCM of 18 and 12?
- A.30
- B.36
- C.42
- D.54
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Answer: B. 36
The smallest number that both 18 and 12 divide into exactly is 36.
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38. Which statement about 226 is certainly true?
- A.226 is divisible by 2
- B.226 is divisible by 5
- C.226 is divisible by 10
- D.226 is divisible by 9
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Answer: A. 226 is divisible by 2
226 matches the statement "226 is divisible by 2" — check the others and they fail.
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39. A number is a perfect number if the sum of its factors (other than itself) equals the number. Which of these is a perfect number?
- A.6
- B.8
- C.10
- D.12
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Answer: A. 6
Factors of 6 (excluding itself): 1, 2, 3 — and 1+2+3 = 6.
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40. Which of the following is NOT a pair of twin primes?
- A.(17, 19)
- B.(29, 31)
- C.(41, 43)
- D.(23, 27)
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Answer: D. (23, 27)
27 = 3×3×3 is not prime, so (23, 27) can't be a twin-prime pair. The other three pairs are both prime and differ by 2.
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41. Which of these numbers is co-prime with 15?
- A.9
- B.10
- C.16
- D.20
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Answer: C. 16
15 = 3×5. 9 shares the factor 3, 10 shares the factor 5, and 20 shares the factor 5 — but 16 = 2⁴ shares nothing with 15.
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42. Find the HCF of 18, 24 and 30.
- A.6
- B.8
- C.9
- D.12
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Answer: A. 6
Common factors of all three are 1, 2, 3, 6 — the highest is 6.
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43. What is the smallest number exactly divisible by both 6 and 8?
- A.16
- B.24
- C.32
- D.48
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Answer: B. 24
Multiples of 6: 6,12,18,24. Multiples of 8: 8,16,24. LCM = 24.
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44. What is the LCM of 9 and 12?
- A.24
- B.36
- C.48
- D.72
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Answer: B. 36
Multiples of 9: 9,18,27,36. Multiples of 12: 12,24,36. LCM is 36.
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45. How many prime numbers are there between 1 and 20?
- A.6
- B.7
- C.8
- D.9
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Answer: C. 8
2, 3, 5, 7, 11, 13, 17, 19 — that's 8 primes.
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46. Which of these numbers is both a perfect square and even?
- A.9
- B.16
- C.25
- D.49
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Answer: B. 16
16 = 4² and is even. 9, 25, and 49 are all odd perfect squares.
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47. What is the smallest number that must be added to 763 to make it divisible by 9?
- A.1
- B.2
- C.3
- D.4
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Answer: B. 2
763 ÷ 9 = 84 remainder 7, so 763 is 7 more than a multiple of 9 — adding 2 more reaches the next multiple: 765 = 9 × 85.
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48. What is the smallest positive number that must be added to 718 to make it divisible by 9?
- A.4
- B.3
- C.1
- D.2
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Answer: D. 2
718 ÷ 9 leaves a remainder, and adding 2 reaches the next multiple of 9.
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49. What is the LCM of 14 and 13?
- A.178
- B.182
- C.195
- D.196
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Answer: B. 182
The smallest number that both 14 and 13 divide into exactly is 182.
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50. What is the remainder when 61 is divided by 7?
- A.6
- B.0
- C.1
- D.5
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Answer: D. 5
61 ÷ 7 leaves a remainder of 5.
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51. What is the smallest positive number that must be added to 488 to make it divisible by 9?
- A.7
- B.9
- C.6
- D.8
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Answer: A. 7
488 ÷ 9 leaves a remainder, and adding 7 reaches the next multiple of 9.
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52. What is the LCM of 5 and 7?
- A.35
- B.31
- C.42
- D.40
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Answer: A. 35
The smallest number that both 5 and 7 divide into exactly is 35.
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53. What is the LCM of 14 and 12?
- A.80
- B.98
- C.84
- D.96
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Answer: C. 84
The smallest number that both 14 and 12 divide into exactly is 84.
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54. What is the smallest positive number that must be added to 214 to make it divisible by 9?
- A.3
- B.4
- C.2
- D.1
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Answer: C. 2
214 ÷ 9 leaves a remainder, and adding 2 reaches the next multiple of 9.
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55. What is the remainder when 190 is divided by 6?
- A.5
- B.0
- C.4
- D.1
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Answer: C. 4
190 ÷ 6 leaves a remainder of 4.
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56. What is the remainder when 116 is divided by 7?
- A.4
- B.6
- C.0
- D.5
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Answer: A. 4
116 ÷ 7 leaves a remainder of 4.
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57. What is the smallest positive number that must be added to 478 to make it divisible by 9?
- A.8
- B.7
- C.10
- D.9
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Answer: A. 8
478 ÷ 9 leaves a remainder, and adding 8 reaches the next multiple of 9.
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58. Which statement about 238 is certainly true?
- A.238 is divisible by 10
- B.238 is divisible by 5
- C.238 is divisible by 9
- D.238 is divisible by 2
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Answer: D. 238 is divisible by 2
238 matches the statement "238 is divisible by 2" — check the others and they fail.
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59. How many factors does 36 have?
- A.11
- B.10
- C.8
- D.9
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Answer: D. 9
36 has 9 factors in total.
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60. Which of the following is a prime number?
- A.49
- B.47
- C.51
- D.45
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Answer: B. 47
47 has no factors other than 1 and itself, so it is prime; the other options can all be divided evenly by a smaller number.
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61. How many factors does 48 have?
- A.10
- B.12
- C.11
- D.9
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Answer: A. 10
48 has 10 factors in total.
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62. Which statement about 182 is certainly true?
- A.182 is divisible by 2
- B.182 is divisible by 10
- C.182 is divisible by 5
- D.182 is divisible by 9
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Answer: A. 182 is divisible by 2
182 matches the statement "182 is divisible by 2" — check the others and they fail.
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63. The next perfect number after 6 is 28 (its factors 1, 2, 4, 7, 14 add up to 28). What is the sum of these two smallest perfect numbers?
- A.28
- B.30
- C.32
- D.34
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Answer: D. 34
6 + 28 = 34.
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64. How many pairs of twin primes are there between 1 and 30?
- A.3
- B.4
- C.5
- D.6
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Answer: B. 4
The primes up to 30 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Twin pairs: (3,5), (5,7), (11,13), (17,19) — 4 pairs.
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65. Which of these is true? Statement I: Every pair of distinct prime numbers is co-prime. Statement II: 1 and any other number are always co-prime.
- A.Only I is true
- B.Only II is true
- C.Both I and II are true
- D.Neither I nor II is true
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Answer: C. Both I and II are true
Two different primes can never share a common factor besides 1, so I is true. And the only factor of 1 is 1 itself, so I is always co-prime with anything — II is also true.
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66. A number between 50 and 60 is prime, and is also one more than a multiple of 4. Which number is it?
- A.51
- B.53
- C.57
- D.59
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Answer: B. 53
53 is prime and 53 = 4×13 + 1. 59 is prime too, but 59−1=58 is not a multiple of 4. 51 and 57 aren't prime at all.
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67. The HCF of two numbers is 6 and their LCM is 72. If one number is 24, what is the other?
- A.12
- B.16
- C.18
- D.20
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Answer: C. 18
HCF × LCM = product of the two numbers: 6 × 72 = 432. 432 ÷ 24 = 18.
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68. A gardener has 48 mango saplings and 60 guava saplings. He wants to plant them in rows with the same number of plants per row (only one type per row), using the maximum possible per row. How many plants will be in each row?
- A.6
- B.8
- C.12
- D.15
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Answer: C. 12
This is the HCF of 48 and 60, which is 12.
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69. Two bells ring together at 9:00 AM. One rings every 12 minutes and the other every 18 minutes. At what time will they next ring together?
- A.9:24 AM
- B.9:30 AM
- C.9:36 AM
- D.9:48 AM
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Answer: C. 9:36 AM
The bells ring together again after the LCM of 12 and 18 minutes, which is 36 minutes — so 9:36 AM.
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70. What is the smallest positive number that must be added to 911 to make it divisible by 9?
- A.9
- B.6
- C.8
- D.7
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Answer: D. 7
911 ÷ 9 leaves a remainder, and adding 7 reaches the next multiple of 9.
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71. What is the remainder when 161 is divided by 6?
- A.2
- B.0
- C.1
- D.5
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Answer: D. 5
161 ÷ 6 leaves a remainder of 5.
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72. What is the smallest positive number that must be added to 243 to make it divisible by 9?
- A.9
- B.8
- C.11
- D.10
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Answer: A. 9
243 is already a multiple of 9 (243 = 27 × 9). Since the number added must be positive, the smallest choice is the next multiple: adding 9 gives 252 = 28 × 9.
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73. What is the smallest positive number that must be added to 698 to make it divisible by 9?
- A.6
- B.3
- C.4
- D.5
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Answer: C. 4
698 ÷ 9 leaves a remainder, and adding 4 reaches the next multiple of 9.
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74. What is the smallest positive number that must be added to 664 to make it divisible by 9?
- A.3
- B.1
- C.4
- D.2
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Answer: D. 2
664 ÷ 9 leaves a remainder, and adding 2 reaches the next multiple of 9.
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75. Which of the following is a prime number?
- A.81
- B.83
- C.85
- D.87
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Answer: B. 83
83 has no factors other than 1 and itself, so it is prime; the other options can all be divided evenly by a smaller number.
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76. What is the HCF of 105 and 90?
- A.15
- B.12
- C.30
- D.18
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Answer: A. 15
The highest number that divides both 105 and 90 exactly is 15.
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77. How many factors does 40 have?
- A.7
- B.8
- C.10
- D.9
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Answer: B. 8
40 has 8 factors in total.
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78. What is the LCM of 24 and 12?
- A.18
- B.48
- C.30
- D.24
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Answer: D. 24
The smallest number that both 24 and 12 divide into exactly is 24.
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79. What is the HCF of 45 and 90?
- A.45
- B.90
- C.42
- D.48
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Answer: A. 45
The highest number that divides both 45 and 90 exactly is 45.
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80. What is the LCM of 54 and 12?
- A.108
- B.114
- C.162
- D.102
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Answer: A. 108
The smallest number that both 54 and 12 divide into exactly is 108.
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81. Which statement about 128 is certainly true?
- A.128 is divisible by 10
- B.128 is divisible by 2
- C.128 is divisible by 5
- D.128 is divisible by 9
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Answer: B. 128 is divisible by 2
128 matches the statement "128 is divisible by 2" — check the others and they fail.
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82. Which statement about 102 is certainly true?
- A.102 is divisible by 2
- B.102 is divisible by 5
- C.102 is divisible by 9
- D.102 is divisible by 10
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Answer: A. 102 is divisible by 2
102 matches the statement "102 is divisible by 2" — check the others and they fail.
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83. Which of the following is a prime number?
- A.93
- B.91
- C.87
- D.89
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Answer: D. 89
89 has no factors other than 1 and itself, so it is prime; the other options can all be divided evenly by a smaller number.
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84. Which statement about 174 is certainly true?
- A.174 is divisible by 2
- B.174 is divisible by 9
- C.174 is divisible by 5
- D.174 is divisible by 10
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Answer: A. 174 is divisible by 2
174 matches the statement "174 is divisible by 2" — check the others and they fail.
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85. How many factors does 30 have?
- A.9
- B.7
- C.10
- D.8
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Answer: D. 8
30 has 8 factors in total.
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86. Two co-prime numbers, both less than 10, have a product of 21. What are they?
- A.3 and 7
- B.4 and 5
- C.2 and 9
- D.1 and 21
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Answer: A. 3 and 7
21 = 3×7, and both 3 and 7 are less than 10 with no common factor. 4×5=20 and 2×9=18 don't give 21, and 21 itself isn't less than 10.
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87. Which of the following numbers is both prime and exactly one less than a perfect square?
- A.3
- B.8
- C.15
- D.24
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Answer: A. 3
3 = 4 − 1 = 2² − 1, and 3 is prime. 8 = 9−1 isn't prime, 15 = 16−1 isn't prime, and 24 = 25−1 isn't prime.
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