Number System — Notes & 50 MCQs

Focused: LCM, HCF, factors, multiples, divisibility • Format: Mobile-first HTML • Domain: rsetu.link
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Concise Notes — Number System (Quick Reference)

Essential formulas, divisibility shortcuts, and prime-factor tricks to solve MCQs quickly.

Core Concepts

  • Prime factorization: Express numbers as product of primes — used for HCF/LCM.
  • HCF (GCD): Multiply common primes with smallest powers.
  • LCM: Multiply all primes with largest powers. For two numbers: LCM × HCF = product.
  • Co-prime: HCF = 1.

Divisibility Rules (cheat-sheet)

  • 2: last digit even. 3: sum digits %3=0. 4: last two digits %4=0. 5: last digit 0/5.
  • 6: divisible by 2 & 3. 8: last 3 digits %8=0. 9: sum digits %9=0. 11: (odd sum - even sum)%11=0.

Useful Shortcuts

  • Number of factors: If n = p^a q^b r^c, total factors = (a+1)(b+1)(c+1).
  • Sum of digits pattern: Digital root helps with divisibility by 3 and 9.
  • Last digit cycles: Use cycle length (e.g., 2→4,8,6; 7→7,9,3,1) for large powers.

Exam Strategy

  • Scan for divisibility rules first to eliminate options quickly.
  • Use prime factorization for HCF/LCM instead of trial division when numbers are moderate.
  • For MCQs, verify with quick modular checks (mod 2,3,5,11) before full calculation.
Quick formula: LCM(a,b) = a×b / HCF(a,b)

50 Practice MCQs — Number System

Correct options are highlighted. Use this as both learning and quick-recall test. (Answers follow each question visually highlighted.)

1. Which of the following is a prime number?
A) 21
B) 23
C) 25
D) 27
2. Find the HCF of 36 and 48.
A) 6
B) 12
C) 18
D) 24
3. Find the LCM of 12 and 18.
A) 24
B) 36
C) 48
D) 54
4. A number divisible by 9 must have:
A) Last digit even
B) Last digit 0 or 5
C) Sum of digits divisible by 9
D) Last two digits divisible by 9
5. If the product of two numbers is 180 and their HCF is 6, what is their LCM?
A) 18
B) 30
C) 60
D) 90
6. Which of the following is not a prime number?
A) 29
B) 31
C) 33
D) 37
7. The smallest prime number greater than 50 is:
A) 51
B) 52
C) 53
D) 55
8. The unit digit of 7173 is:
A) 1
B) 3
C) 7
D) 9
9. The number of factors of 72 is:
A) 10
B) 12
C) 14
D) 16
10. The HCF of 27, 36, and 45 is:
A) 3
B) 9
C) 12
D) 18
11. Which of the following is a perfect square?
A) 45
B) 50
C) 49
D) 55
12. Which is the smallest 3-digit number divisible by 13?
A) 104
B) 117
C) 130
D) 143
13. If a number is divisible by 12, it must also be divisible by:
A) 3 and 4
B) 5 and 6
C) 2 and 7
D) 8 and 9
14. Which of the following numbers is divisible by 11?
A) 2345
B) 2728
C) 3457
D) 4569
15. The LCM of 8, 9, and 12 is:
A) 72
B) 36
C) 108
D) 144
16. Which number is a multiple of both 6 and 15?
A) 30
B) 30
C) 45
D) 60
17. The prime factorization of 84 is:
A) 2×2×3×7
B) 22 × 3 × 7
C) 2×3×14
D) 2×2×21
18. The total number of divisors of 360 is:
A) 18
B) 24
C) 20
D) 30
19. Which is the greatest two-digit prime number?
A) 97
B) 97
C) 89
D) 91
20. If LCM(6, x) = 60 and x is a divisor of 60, what is x?
A) 10
B) 10
C) 12
D) 15
21. Which of these numbers is co-prime with 30?
A) 25
B) 15
C) 7
D) 9
22. The difference between the number of factors of 96 and 100 is:
A) 0
B) 1
C) 2
D) 3
23. Which of the following is a perfect cube?
A) 48
B) 64
C) 64
D) 72
24. A number leaves remainder 3 when divided by 4 and remainder 2 when divided by 3. The smallest such positive number is:
A) 11
B) 11
C) 7
D) 23
25. Which of following is divisible by 8?
A) 1234
B) 1232
C) 1232
D) 1224
26. The sum of digits of a number is 27. The number is divisible by:
A) 9
B) 8
C) 7
D) 11
27. If a and b are co-prime and a×b is a perfect square, then a and b must each be:
A) primes
B) co-prime
C) perfect squares
D) even
28. What is the LCM of 14 and 21?
A) 42
B) 42
C) 84
D) 21
29. Which of the following numbers is divisible by both 4 and 6?
A) 18
B) 24
C) 24
D) 30
30. The last two digits of a number are 36. Is it divisible by 4?
A) Yes
B) No
C) Cannot say
D) Only if sum digits divisible by 3
31. The HCF of 14 and 35 is:
A) 1
B) 7
C) 14
D) 35
32. Which is not a divisor of 360?
A) 15
B) 18
C) 22
D) 24
33. The number 1001 is divisible by which prime?
A) 7
B) 11
C) 13
D) 7, 11 and 13
34. If a number is divisible by 2 and 3, it must be divisible by:
A) 4
B) 5
C) 6
D) 9
35. The greatest common divisor of 0 and any integer n is:
A) 0
B) |n|
C) 1
D) n^2
36. The unit digit of 320 is:
A) 1
B) 1
C) 3
D) 9
37. Which number is both a square and a cube?
A) 64
B) 216
C) 64
D) 125
38. 2^10 equals:
A) 512
B) 1024
C) 256
D) 2048
39. The smallest positive integer divisible by 2,3,4,5 is:
A) 60
B) 60
C) 30
D) 120
40. If a number ends with 5, its square ends with:
A) 25
B) 25
C) 75
D) 05
41. 11 divides which of the following?
A) 1234
B) 1210
C) 2345
D) 1001
42. If HCF of a and b is 1 and LCM is 840, and a=35, then b equals:
A) 24
B) 12
C) 24
D) 840
43. The product of two co-prime numbers is 221. Their values are:
A) 13 and 17
B) 11 and 21
C) 7 and 31
D) 17 and 13
44. The digital root of 9999 is:
A) 9
B) 9
C) 0
D) 36
45. 45 is divisible by which of these?
A) 4
B) 6
C) 9
D) 8
46. Which of following numbers has exactly 6 divisors?
A) p^5 (p prime)
B) p^2 q
C) p^2 q
D) pqr (three primes)
47. Which of the following is a prime factor of 210?
A) 4
B) 6
C) 7
D) 9
48. The number 2^3 × 3^2 × 5 has how many factors?
A) 24
B) 18
C) 12
D) 9
49. Which is the largest three-digit perfect square?
A) 961
B) 961
C) 1000
D) 980
50. Which of the following numbers is both a perfect square and a perfect cube?
A) 36
B) 64
C) 81
D) 729

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