Worksheet - Factors and Multiples

Total Marks: / 40 marks

Booklet A - Multiple Choice Questions (MCQ)

Each question carries 1 mark. Circle the letter of the correct answer.

Q1: Which of the following numbers is a factor of 36?

1. 5
2. 7
3. 9
4. 10

Q2: What is the smallest common multiple of 4 and 6?

1. 12
2. 24
3. 2
4. 6

Q3: How many factors does 28 have?

1. 4
2. 5
3. 6
4. 7

Q4: Which number below is NOT a multiple of 8?

1. 32
2. 48
3. 56
4. 60

Q5: What is the greatest common factor of 24 and 36?

1. 6
2. 12
3. 18
4. 72

Q6: A number has exactly two factors. What type of number is it?

1. Composite number
2. Prime number
3. Even number
4. Square number

Q7: The highest common factor of two numbers is 8. The numbers could be:

1. 16 and 20
2. 24 and 40
3. 12 and 18
4. 10 and 15

Q8: Ali wrote down all the common multiples of 6 and 9 between 30 and 100. How many numbers did he write?

1. 3
2. 4
3. 5
4. 6

Booklet B - Short Answer Questions (SAQ)

Show your working clearly. Questions carry 2 marks unless stated.

Q9: List all the factors of 42.

Q10: Find the lowest common multiple of 12 and 15.

Q11: What is the greatest common factor of 48 and 72?

Q12: A number is both a multiple of 7 and a factor of 84. List all the possible numbers.

Q13: The product of two prime numbers is 51. What are the two prime numbers?

Q14: Two bells ring at regular intervals. One bell rings every 8 minutes and the other rings every 12 minutes. If both bells ring together at 9:00 a.m., at what time will they next ring together?

Q15: Express 60 as a product of its prime factors.

Q16: Mrs Tan wants to arrange 48 chocolates and 72 sweets into identical gift bags with no items left over. What is the greatest number of gift bags she can prepare? How many chocolates and how many sweets will be in each bag? [3 marks]

Booklet B - Long Answer / Problem Sums (LAQ)

Show all working clearly. Marks are awarded for method and final answer.

Q17: Three lights flash at intervals of 6 seconds, 8 seconds and 12 seconds respectively. If all three lights flash together at 7:15 p.m., find: [4 marks]

(a) The time when all three lights will next flash together.
(b) How many times all three lights will flash together between 7:15 p.m. and 7:20 p.m., including the flash at 7:15 p.m.

Q18: A fruit seller packs mangoes into boxes of 15 and oranges into boxes of 18. He finds that he has packed an equal number of mangoes and oranges. [4 marks]

(a) What is the smallest possible number of mangoes he has packed?
(b) How many boxes of each fruit did he pack?

Q19: At a hawker centre, plates are delivered every 20 days, bowls every 30 days, and cups every 24 days. On 1 March, all three items were delivered together. [5 marks]

(a) On what date will all three items next be delivered together?
(b) Between 1 March and 1 September of the same year, how many times will plates and bowls be delivered together (including 1 March)?

Q20: The product of the highest common factor and the lowest common multiple of two numbers is 360. One of the numbers is 12. [5 marks]

(a) Find the other number.
(b) If both numbers are less than 50, how many possible pairs of such numbers are there? List all the possible pairs.

Answer Key

Multiple Choice Questions

MCQ Explanations

Q1:

Ans: C
Explanation: The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36. Among the options only 9 is in this list because 36 ÷ 9 = 4 with no remainder. Therefore (C) is correct.

Q2:

Ans: A
Explanation: To find the smallest common multiple (LCM) of 4 and 6, list a few multiples:
Multiples of 4: 4, 8, 12, 16, 20...
Multiples of 6: 6, 12, 18, 24...
The first common multiple is 12. Hence (A) is correct.

Q3:

Ans: C
Explanation: The factors of 28 are 1, 2, 4, 7, 14 and 28. Counting them gives 6 factors in total. Hence (C) is correct.

Q4:

Ans: D
Explanation: Check divisibility by 8:
32 ÷ 8 = 4, 48 ÷ 8 = 6, 56 ÷ 8 = 7, but 60 ÷ 8 = 7.5 which is not an integer. Therefore 60 is not a multiple of 8 and (D) is correct.

Q5:

Ans: B
Explanation: Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Common factors are 1, 2, 3, 4, 6 and 12. The greatest common factor is 12. So (B) is correct.

Q6:

Ans: B
Explanation: A prime number has exactly two distinct factors: 1 and the number itself. For example, 7 has factors 1 and 7. Thus (B) is correct.

Q7:

Ans: B
Explanation: Find the highest common factor for each pair:
A: HCF(16,20) = 4
B: HCF(24,40) = 8 ✓
C: HCF(12,18) = 6
D: HCF(10,15) = 5
Only pair B has HCF 8, so (B) is correct.

Q8:

Ans: B
Explanation: The LCM of 6 and 9 is 18. Common multiples between 30 and 100 that are multiples of 18 are 36, 54, 72 and 90. There are 4 numbers, so (B) is correct.

Short Answer Questions

Q9: List all the factors of 42.

Ans: 1, 2, 3, 6, 7, 14, 21, 42
Working:
Pairs that multiply to 42 are:
1 × 42 = 42
2 × 21 = 42
3 × 14 = 42
6 × 7 = 42
Collecting all factors gives 1, 2, 3, 6, 7, 14, 21, 42.

Q10: Find the lowest common multiple of 12 and 15.

Ans: 60
Working:
Use prime factors:
12 = 2^2 × 3
15 = 3 × 5
LCM = 2^2 × 3 × 5 = 60.

Q11: What is the greatest common factor of 48 and 72?

Ans: 24
Working:
Prime factorise:
48 = 2^4 × 3
72 = 2^3 × 3^2
Take the lowest powers of common primes: 2^3 × 3 = 8 × 3 = 24.
So HCF = 24.

Q12: A number is both a multiple of 7 and a factor of 84. List all the possible numbers.

Ans: 7, 14, 21, 28, 42, 84
Working:
Factors of 84 include 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84.
From these, those that are multiples of 7 are 7, 14, 21, 28, 42 and 84.

Q13: The product of two prime numbers is 51. What are the two prime numbers?

Ans: 3 and 17
Working:
51 = 3 × 17 and both 3 and 17 are prime numbers. Therefore the two primes are 3 and 17.

Q14: Two bells ring at regular intervals. One bell rings every 8 minutes and the other rings every 12 minutes. If both bells ring together at 9:00 a.m., at what time will they next ring together?

Ans: 9:24 a.m.
Working:
Find the LCM of 8 and 12.
8 = 2^3, 12 = 2^2 × 3 → LCM = 2^3 × 3 = 24 minutes.
Next time = 9:00 a.m. + 24 minutes = 9:24 a.m.

Q15: Express 60 as a product of its prime factors.

Ans: 2 × 2 × 3 × 5 or 2^2 × 3 × 5
Working:
60 = 2 × 30 = 2 × 2 × 15 = 2 × 2 × 3 × 5.

Q16: Mrs Tan wants to arrange 48 chocolates and 72 sweets into identical gift bags with no items left over. What is the greatest number of gift bags she can prepare? How many chocolates and how many sweets will be in each bag?

Ans: 24 gift bags; 2 chocolates and 3 sweets in each bag
Working:
Find the HCF of 48 and 72 using prime factors:
48 = 2^4 × 3
72 = 2^3 × 3^2
HCF = 2^3 × 3 = 8 × 3 = 24.
Number of gift bags = 24.
Chocolates per bag = 48 ÷ 24 = 2.
Sweets per bag = 72 ÷ 24 = 3.

Long Answer / Problem Sums

Q17: Three lights flash at intervals of 6 seconds, 8 seconds and 12 seconds respectively. If all three lights flash together at 7:15 p.m., find:

(a) The time when all three lights will next flash together.

(b) How many times all three lights will flash together between 7:15 p.m. and 7:20 p.m., including the flash at 7:15 p.m.

Ans:
(a) 7:15:24 p.m.
(b) 13 times
Working:
Find the LCM of 6, 8 and 12 by prime factors:
6 = 2 × 3, 8 = 2^3, 12 = 2^2 × 3 → LCM = 2^3 × 3 = 8 × 3 = 24 seconds.
(a) Next time = 7:15:00 p.m. + 24 seconds = 7:15:24 p.m.
(b) Time interval = 7:15:00 p.m. to 7:20:00 p.m. = 5 minutes = 300 seconds.
Number of intervals of 24 seconds in 300 seconds = floor(300 ÷ 24) = 12 complete intervals after the first flash.
Including the initial flash at 7:15:00, total flashes together = 12 + 1 = 13 times.

Q18: A fruit seller packs mangoes into boxes of 15 and oranges into boxes of 18. He finds that he has packed an equal number of mangoes and oranges.

(a) What is the smallest possible number of mangoes he has packed?

(b) How many boxes of each fruit did he pack?

Ans:
(a) 90 mangoes
(b) 6 boxes of mangoes and 5 boxes of oranges
Working:
Find the LCM of 15 and 18:
15 = 3 × 5, 18 = 2 × 3^2 → LCM = 2 × 3^2 × 5 = 90.
(a) Smallest equal number = 90.
(b) Boxes of mangoes = 90 ÷ 15 = 6 boxes.
Boxes of oranges = 90 ÷ 18 = 5 boxes.

Q19: At a hawker centre, plates are delivered every 20 days, bowls every 30 days, and cups every 24 days. On 1 March, all three items were delivered together.

(a) On what date will all three items next be delivered together?

(b) Between 1 March and 1 September of the same year, how many times will plates and bowls be delivered together (including 1 March)?

Ans:
(a) 29 June
(b) 4 times
Working:
(a) Use prime factors to find LCM of 20, 30 and 24:
20 = 2^2 × 5, 30 = 2 × 3 × 5, 24 = 2^3 × 3 → LCM = 2^3 × 3 × 5 = 8 × 3 × 5 = 120 days.
Next common delivery = 1 March + 120 days = 29 June.
(b) For plates and bowls, find LCM of 20 and 30:
20 = 2^2 × 5, 30 = 2 × 3 × 5 → LCM = 2^2 × 3 × 5 = 60 days.
Count deliveries from 1 March up to 1 September. The period from 1 March to 1 September is 184 days.
Number of times plates and bowls are delivered together = floor(184 ÷ 60) + 1 (including 1 March) = 3 + 1 = 4 times.
These occur on 1 March, 30 April, 29 June and 28 August.

Q20: The product of the highest common factor and the lowest common multiple of two numbers is 360. One of the numbers is 12.

(a) Find the other number.

(b) If both numbers are less than 50, how many possible pairs of such numbers are there? List all the possible pairs.

Ans:
(a) 30
(b) 6 pairs: (8, 45), (9, 40), (10, 36), (12, 30), (15, 24), (18, 20)
Working:
Use the identity: For any two positive integers a and b,
HCF(a, b) × LCM(a, b) = a × b.
(a) Let the other number be n. Given HCF × LCM = 360 and one number is 12, so 12 × n = 360. Thus n = 360 ÷ 12 = 30.
(b) The condition HCF × LCM = 360 is equivalent to a × b = 360. We list factor pairs of 360 and select those where both numbers are less than 50:
Factor pairs of 360: (1, 360), (2, 180), (3, 120), (4, 90), (5, 72), (6, 60), (8, 45), (9, 40), (10, 36), (12, 30), (15, 24), (18, 20).

Therefore there are 6 such pairs (listed above). Note these are unordered pairs; reversing any pair gives the same two numbers.