About this quiz
Number system, HCF, LCM and divisibility.
This Lok Sewa practice set has 50 multiple-choice questions (Subject: Maths and IQ · For: Kharidar, Nayab Subba, Section Officer · Level: Medium). Take the quiz above first; after that, use the study list below to revise every question with its correct answer and a short explanation.
Study list: all 50 questions with answers and explanations
Which is the only even prime number?
- A) 0
- B) 2 ✔
- C) 4
- D) 6
Why: 2 has only two factors, 1 and 2, and is even.
How many prime numbers are there between 1 and 100?
- A) 21
- B) 24
- C) 25 ✔
- D) 26
Why: There are 25 primes up to 100.
The number 1 is:
- A) Prime
- B) Composite
- C) Neither prime nor composite ✔
- D) Both prime and composite
Why: 1 has only one factor.
√2 is a/an:
- A) Rational number
- B) Irrational number ✔
- C) Integer
- D) Natural number
Why: √2 cannot be written as p/q.
Two numbers are co-prime if their HCF is:
- A) 0
- B) 1 ✔
- C) 2
- D) Equal to their LCM
Why: Co-prime numbers have no common factor except 1.
HCF of 36 and 48 is:
- A) 6
- B) 12 ✔
- C) 18
- D) 24
Why: 36 = 2²×3², 48 = 2⁴×3; lowest common powers 2²×3 = 12.
LCM of 36 and 48 is:
- A) 96
- B) 144 ✔
- C) 288
- D) 72
Why: Highest powers 2⁴×3² = 144.
LCM of 12, 18 and 24 is:
- A) 36
- B) 48
- C) 72 ✔
- D) 144
Why: 2³×3² = 72.
The HCF of two numbers is 8 and their LCM is 96. If one number is 24, the other is:
- A) 28
- B) 32 ✔
- C) 36
- D) 40
Why: 8×96 ÷ 24 = 32.
Which number is divisible by 9?
- A) 3,481
- B) 5,623
- C) 4,572 ✔
- D) 7,201
Why: 4+5+7+2 = 18, which is divisible by 9.
Which number is divisible by 11?
- A) 4,321
- B) 5,918 ✔
- C) 7,523
- D) 6,132
Why: Rule for 11: take the alternating sum of the digits. For 5,918: 5 − 9 + 1 − 8 = −11, which is a multiple of 11, so 5,918 is divisible by 11 (5,918 = 11 × 538).
A number is divisible by 8 if:
- A) Its last digit is divisible by 8
- B) Its last two digits are divisible by 8
- C) Its last three digits are divisible by 8 ✔
- D) The sum of its digits is divisible by 8
Why: Divisibility by 8 depends on the last three digits.
A number is divisible by 6 if it is divisible by:
- A) 2 and 3 ✔
- B) 3 and 4
- C) 2 and 4
- D) 6 and 12
Why: 6 = 2×3, with 2 and 3 co-prime.
Three bells ring every 6, 8 and 12 minutes. If they ring together now, after how many minutes will they ring together again?
- A) 12
- B) 24 ✔
- C) 36
- D) 48
Why: LCM of 6, 8 and 12 is 24.
The smallest number which, when divided by 12, 15 and 20, leaves remainder 5 in each case is:
- A) 60
- B) 65 ✔
- C) 125
- D) 55
Why: LCM 60 + 5 = 65.
The largest number that divides 43 and 91 leaving remainder 7 in each case is:
- A) 6
- B) 12 ✔
- C) 14
- D) 18
Why: HCF of 36 and 84 is 12.
The smallest 4-digit number exactly divisible by 12, 15 and 18 is:
- A) 1,000
- B) 1,080 ✔
- C) 1,180
- D) 1,260
Why: LCM is 180; 180×6 = 1,080.
Two numbers are in the ratio 3:4 and their HCF is 5. Their LCM is:
- A) 20
- B) 35
- C) 60 ✔
- D) 120
Why: Numbers are 15 and 20; LCM = 60.
The sum of the first 50 natural numbers is:
- A) 1,225
- B) 1,275 ✔
- C) 2,500
- D) 2,550
Why: 50×51 ÷ 2 = 1,275.
The sum of the first 20 odd numbers is:
- A) 200
- B) 210
- C) 400 ✔
- D) 420
Why: Sum of first n odd numbers = n² = 400.
HCF of 12 and 18 is:
- A) 6 ✔
- B) 3
- C) 36
- D) 12
Why: 12 = 2² × 3 and 18 = 2 × 3². HCF takes the lowest common powers: 2 × 3 = 6. 6 is the largest number that divides both.
LCM of 12 and 18 is:
- A) 36 ✔
- B) 72
- C) 6
- D) 216
Why: 12 = 2² × 3 and 18 = 2 × 3². LCM takes the highest powers: 2² × 3² = 36. Check: 36 ÷ 12 = 3 and 36 ÷ 18 = 2.
For two numbers, HCF × LCM equals:
- A) Product of the numbers ✔
- B) Sum of the numbers
- C) Difference
- D) Their ratio
Why: For any two numbers, HCF × LCM = product of the numbers. Example: 12 and 18 have HCF 6 and LCM 36, and 6 × 36 = 216 = 12 × 18.
HCF of 24, 36 and 60 is:
- A) 12 ✔
- B) 6
- C) 24
- D) 4
Why: 24 = 2³ × 3, 36 = 2² × 3², 60 = 2² × 3 × 5. The common part is 2² × 3 = 12, so the HCF is 12.
LCM of 4, 6 and 8 is:
- A) 24 ✔
- B) 48
- C) 12
- D) 16
Why: 4 = 2², 6 = 2 × 3, 8 = 2³. LCM = 2³ × 3 = 24, the smallest number all three divide.
How many prime numbers are there from 1 to 100?
- A) 25 ✔
- B) 24
- C) 26
- D) 20
Why: There are 25 primes from 1 to 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
Which number is divisible by 3?
- A) 4,512 ✔
- B) 4,513
- C) 2,221
- D) 1,001
Why: A number is divisible by 3 if its digit sum is. For 4,512: 4 + 5 + 1 + 2 = 12, and 12 ÷ 3 = 4, so 4,512 is divisible by 3.
Which number is divisible by 4?
- A) 3,716 ✔
- B) 3,714
- C) 2,522
- D) 5,610
Why: A number is divisible by 4 if its last two digits are. 3,716 ends in 16, and 16 ÷ 4 = 4, so 3,716 is divisible by 4.
Which number is divisible by 11?
- A) 2,728 ✔
- B) 3,456
- C) 1,234
- D) 9,875
Why: For 2,728: (2 + 2) − (7 + 8) = 4 − 15 = −11, a multiple of 11, so 2,728 is divisible by 11 (2,728 = 11 × 248).
Which is a perfect number?
- A) 28 ✔
- B) 24
- C) 30
- D) 20
Why: A perfect number equals the sum of its factors other than itself. 28: 1 + 2 + 4 + 7 + 14 = 28. The smallest perfect number is 6 (1 + 2 + 3).
Which is an irrational number?
- A) √3 ✔
- B) √9
- C) 0.5
- D) 3/4
Why: √3 is irrational: its decimal (1.732…) never ends and never repeats, so it cannot be written as p/q. Square roots of non-perfect squares are irrational.
0.333... as a fraction is:
- A) 1/3 ✔
- B) 3/10
- C) 33/100
- D) 1/30
Why: Let x = 0.333… Then 10x = 3.333…, so 10x − x = 3, 9x = 3 and x = 3/9 = 1/3.
Sum of the first 10 natural numbers is:
- A) 55 ✔
- B) 50
- C) 45
- D) 100
Why: Use n(n + 1)/2: 10 × 11 ÷ 2 = 55. So 1 + 2 + … + 10 = 55.
Sum of the first n odd numbers is:
- A) n² ✔
- B) n(n+1)
- C) 2n
- D) n³
Why: The first n odd numbers add up to n². Examples: 1 + 3 = 4 = 2², and 1 + 3 + 5 = 9 = 3².
The smallest composite number is:
- A) 4 ✔
- B) 1
- C) 2
- D) 6
Why: 4 is the smallest composite number: its factors are 1, 2 and 4. 1 is neither prime nor composite, and 2 and 3 are prime.
Which pair are twin primes?
- A) 11 and 13 ✔
- B) 13 and 17
- C) 7 and 11
- D) 15 and 17
Why: Twin primes are two primes that differ by 2. 11 and 13 are both prime and 13 − 11 = 2. Other pairs: (3, 5), (5, 7), (17, 19).
How many factors does 36 have?
- A) 9 ✔
- B) 8
- C) 6
- D) 12
Why: 36 = 2² × 3², so the number of factors is (2 + 1)(2 + 1) = 9: 1, 2, 3, 4, 6, 9, 12, 18, 36.
The unit digit of 7⁴ is:
- A) 1 ✔
- B) 7
- C) 9
- D) 3
Why: Unit digits of powers of 7 repeat in a cycle of 4: 7, 9, 3, 1. So 7⁴ ends in 1 (7⁴ = 2,401).
The unit digit of 2¹⁰ is:
- A) 4 ✔
- B) 2
- C) 8
- D) 6
Why: Unit digits of powers of 2 repeat as 2, 4, 8, 6. 10 ÷ 4 leaves remainder 2, so the unit digit is the 2nd in the cycle, 4 (2¹⁰ = 1,024).
The remainder when 100 is divided by 7 is:
- A) 2 ✔
- B) 3
- C) 1
- D) 5
Why: 7 × 14 = 98, and 100 − 98 = 2, so the remainder is 2.
LCM of co-prime numbers 7 and 9 is:
- A) 63 ✔
- B) 1
- C) 16
- D) 9
Why: Co-prime numbers have HCF 1, so their LCM is their product: 7 × 9 = 63.
HCF of two consecutive numbers is always:
- A) 1 ✔
- B) 2
- C) 0
- D) The smaller number
Why: Any common factor of n and n + 1 must also divide their difference, 1. So consecutive numbers are co-prime and their HCF is always 1.
The largest number that divides both 25 and 40 exactly is:
- A) 5 ✔
- B) 10
- C) 15
- D) 25
Why: This is the HCF. 25 = 5² and 40 = 2³ × 5, so the only common factor is 5.
The smallest number divisible by 2, 3, 4, 5 and 6 is:
- A) 60 ✔
- B) 120
- C) 30
- D) 90
Why: This is the LCM: 2² × 3 × 5 = 60. 4 covers 2, and 6 is covered by 4 and 3.
A rational number between 1/2 and 1 is:
- A) 3/4 ✔
- B) 1/3
- C) 5/4
- D) 1/4
Why: 1/2 = 0.5 and 1 = 1.0, and 3/4 = 0.75 lies between them. A quick method: the average of the two, (1/2 + 1) ÷ 2 = 3/4.
The product of two numbers is 192 and their HCF is 4. Their LCM is:
- A) 48 ✔
- B) 768
- C) 24
- D) 12
Why: HCF × LCM = product, so LCM = 192 ÷ 4 = 48.
How many even numbers are there from 1 to 21?
- A) 10 ✔
- B) 11
- C) 9
- D) 20
Why: The even numbers are 2, 4, 6, …, 20. That is 20 ÷ 2 = 10 numbers.
Which number is divisible by both 2 and 3 (i.e. by 6)?
- A) 114 ✔
- B) 115
- C) 116
- D) 118
Why: 114 is even (divisible by 2) and its digit sum 1 + 1 + 4 = 6 is divisible by 3, so it is divisible by 6 (114 = 6 × 19).
A number is divisible by 25 if its last two digits are:
- A) 00, 25, 50 or 75 ✔
- B) Even
- C) Odd
- D) Zero only
Why: Every multiple of 100 is divisible by 25, so only the last two digits matter. They must be 00, 25, 50 or 75, as in 1,350 or 2,475.
Which is the smallest 3-digit prime number?
- A) 101 ✔
- B) 100
- C) 103
- D) 107
Why: 101 is prime: it is not divisible by 2, 3, 5 or 7, and √101 is about 10. So 101 is the smallest 3-digit prime (100 is composite).