Factors and Multiples Class 6 ICSE — Chapter 4
Factors and multiples class 6 ICSE maths starts with one simple question: which natural numbers divide a given number completely? Chapter 4 of your Class 6 Mathematics book answers it through factors, prime numbers and prime factors, then defines H.C.F. and L.C.M. and gives three methods for finding each. The sections below follow the book’s own order and numbering, with a worked example for every method. Browse the other Class 6 Maths chapters here.
Download Factors and Multiples Class 6 ICSE Mathematics Chapter 4 PDF
Download the Factors and Multiples Class 6 ICSE Mathematics chapter PDF to revise offline or print for practice. It is the full chapter as printed in the book — 11 pages, 15 sections, 24 questions.
What is inside Chapter 4
11 pages · 15 sections · 18 worked examples · 24 exercise questions
| Section | Title |
|---|---|
| 4.1 | FACTORS |
| 4.2 | PRIME NUMBERS |
| 4.3 | PRIME NUMBERS FROM 1 TO 100 |
| 4.4 | PRIME FACTORS |
| 4.5 | HIGHEST COMMON FACTOR |
| 4.6 | METHODS OF FINDING H.C.F. |
| 4.7 | COMMON FACTOR METHOD |
| 4.8 | PRIME FACTOR METHOD |
| 4.9 | DIVISION METHOD |
| 4.10 | MULTIPLES |
| 4.11 | LOWEST COMMON MULTIPLE |
| 4.12 | METHODS OF FINDING L.C.M. |
| 4.13 | COMMON MULTIPLE METHOD |
| 4.14 | PRIME FACTOR METHOD |
| 4.15 | COMMON DIVISION METHOD |
- EXERCISE 4(A): 0 questions
- EXERCISE 4(B): 8 questions
- EXERCISE 4(C): 8 questions
- Revision Exercise (Chapter 4): 8 questions
4.1 FACTORS
A factor is a natural number that divides a given natural number completely. When you multiply numbers together, each one is a factor of the product. For example, 5 \times 7 = 35, so 5 and 7 are factors of 35.
The book writes factors with the letter F. To list all factors, pair the number up: 20 = 1 \times 20 = 2 \times 10 = 4 \times 5. So F_{20} = 1, 2, 4, 5, 10, 20. Two rules always hold: 1 is a factor of every number, and every number is a factor of itself.
4.2 PRIME NUMBERS
A prime number is a natural number divisible only by 1 and itself. Test a number by counting its factors. Exactly two factors means prime; more than two means composite. The number 11 has factors 1 and 11 only, so it is prime. The number 15 has factors 1, 3, 5 and 15, so it is composite. The number 23 is prime.
Three properties matter in this chapter:
- Every prime number is greater than 1.
- 2 is the smallest prime number.
- All primes except 2 are odd, and every even number greater than 2 is composite.
Why is 1 not prime? A prime must have exactly two factors, but 1 has only one factor — itself. That is the most common error students make here.
4.3 PRIME NUMBERS FROM 1 TO 100
Eratosthenes, a Greek scholar, built a method to separate primes from the rest of the natural numbers. It is called the Sieve of Eratosthenes. Write the numbers 1 to 100 in rows of ten. Strike out 1. Encircle 2 and strike out every multiple of 2. Encircle 3 and strike its multiples — some are already cut. Do the same for 5, then 7, then 11, and continue. Numbers already struck by an earlier prime are not struck twice.
When nothing is left to cross, the encircled numbers are the 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.
4.4 PRIME FACTORS
A prime factor is a factor that is also a prime number. The book uses two notations side by side: F_n for all factors and P.F_n for prime factors only. Take 42. Its factors are F_{42} = 1, 2, 3, 6, 7, 14, 21, 42. Among these, 2, 3 and 7 are primes, so P.F_{42} = 2, 3, 7.
Exercise 4(A) questions 4 and 5 ask you to write prime factors and to find values like P_6 and P_{42}. In each case, list the factors first, then keep only the prime ones.
4.5 HIGHEST COMMON FACTOR
H.C.F. stands for Highest Common Factor. The H.C.F. of two or more numbers is the greatest number that divides each of them completely. For example, the largest number that divides both 20 and 30 is 10, so the H.C.F. of 20 and 30 = 10. You can check: 10 divides 20 and 30 exactly, and no larger number does. This is the key term of the chapter — every later method is built to find it.
4.6 METHODS OF FINDING H.C.F.
The book teaches three methods for finding the H.C.F. of two or more numbers. You can choose whichever fits the numbers you are given.
- Common Factor Method — list all factors, pick the common ones, take the highest.
- Prime Factor Method — split into prime factors, multiply the common primes.
- Division Method — divide repeatedly until no remainder is left; the last divisor is the H.C.F.
| Method | Best when | Why |
|---|---|---|
| Common Factor | Small numbers | Factors are few and quick to list. |
| Prime Factor | Medium numbers | Prime splitting is fast and neat. |
| Division | Large numbers | No long factor lists are needed. |
4.7 COMMON FACTOR METHOD
The common factor method has three steps. First, list all factors of each number. Second, select the factors that appear in every list. Third, take the highest of those common factors — that is the H.C.F.
Find the H.C.F. of 30 and 45. F_{30} = 1, 2, 3, 5, 6, 10, 15, 30 and F_{45} = 1, 3, 5, 9, 15, 45. The common factors are 1, 3, 5 and 15. The highest is 15, so the H.C.F. of 30 and 45 = 15.
One edge case: when the only common factor is 1, the H.C.F. is 1. For instance, 16, 32 and 49 share only the factor 1, so their H.C.F. is 1.
4.8 PRIME FACTOR METHOD
The prime factor method also has three steps. Split each number into prime factors. Select the prime factors common to all the numbers. Multiply them — the product is the H.C.F.
Find the H.C.F. of 40 and 60. 40 = 2 \times 2 \times 2 \times 5 and 60 = 2 \times 2 \times 3 \times 5. The common primes are 2, 2 and 5. Multiply: 2 \times 2 \times 5 = 20. So the H.C.F. of 40 and 60 = 20.
This section also defines co-prime numbers — two numbers with no common prime factor, such as 39 and 175. The H.C.F. of two co-prime numbers is always 1.
4.9 DIVISION METHOD
The division method suits large numbers, where listing factors is slow. Divide the greater number by the smaller. Divide the smaller by the remainder. Then divide that remainder by the next remainder. Continue until no remainder is left — the last divisor is the H.C.F.
Find the H.C.F. of 42 and 56. 56 \div 42 = 1 with remainder 14. Then 42 \div 14 = 3 with no remainder. The last divisor is 14, so the H.C.F. of 42 and 56 = 14.
For three or more numbers, first find the H.C.F. of any two. Then find the H.C.F. of that result and the third number, and carry on.
4.10 MULTIPLES
Multiples of a number are its products with 1, 2, 3 and so on. The book writes them as M_n. For example, M_6 = 6, 12, 18, 24, 30, 36, \dots. Unlike factors, multiples never end — you can always multiply by a larger natural number. A factor divides the number; a multiple is divided by the number. That one sentence tells the two ideas apart in this ICSE chapter.
4.11 LOWEST COMMON MULTIPLE
L.C.M. stands for Lowest Common Multiple. The L.C.M. of two or more numbers is the smallest number that is exactly divisible by each of them. For example, multiples of 12 are 12, 24, 36, 48,… and multiples of 18 are 18, 36, 54,…. The smallest common multiple is 36, so the L.C.M. of 12 and 18 = 36. The next three sections give the methods to find it.
4.12 METHODS OF FINDING L.C.M.
The book teaches three methods for finding the L.C.M. of two or more numbers.
- Common Multiple Method — list multiples, pick the lowest common one.
- Prime Factor Method — take the highest power of each prime factor.
- Common Division Method — divide by primes in a table until the bottom row is prime to one another.
For small numbers the common multiple method is quick. For larger numbers the prime factor and common division methods are faster.
4.13 COMMON MULTIPLE METHOD
The common multiple method has three steps. List a few multiples of each number. Select the multiples that appear in every list. Take the lowest of them — that is the L.C.M.
Find the L.C.M. of 6 and 8. M_6 = 6, 12, 18, 24, 30, \dots and M_8 = 8, 16, 24, 32, \dots. The common multiples are 24, 48,…. The lowest is 24, so the L.C.M. of 6 and 8 = 24.
Common multiples repeat at regular intervals, so the smallest one is always the L.C.M.
4.14 PRIME FACTOR METHOD
This method uses index form. First express each number as a product of prime factors, with repeated primes written as powers. Then take the highest power of each prime that appears and multiply them.
Find the L.C.M. of 16 and 24. 16 = 2^4 and 24 = 2^3 \times 3. The prime 2 appears up to power 4, and 3 up to power 1. So L.C.M. = 2^4 \times 3 = 16 \times 3 = 48.
The highest power of each prime matters — not the common ones, as in the H.C.F. method.
4.15 COMMON DIVISION METHOD
The common division method builds a table. Write the numbers in a row. Divide by a prime that divides at least two of them, writing quotients and undivided numbers below. Repeat until the bottom row is prime to one another — no two share a factor. Multiply all divisors and the bottom row: that product is the L.C.M.
Find the L.C.M. of 12, 18 and 24. Divide by 2: 12, 18, 24 \rightarrow 6, 9, 12. Divide by 3: 6, 9, 12 \rightarrow 2, 3, 4. Divide by 2: 2, 3, 4 \rightarrow 1, 3, 2. The bottom row 1, 3, 2 is prime to one another. L.C.M. = 2 \times 3 \times 2 \times 1 \times 3 \times 2 = 72.
The section ends with an important result: for any two numbers,
H.C.F. \times L.C.M. = \text{product of the two numbers}
Take 12 and 18. Their H.C.F. is 6 and L.C.M. is 36. 6 \times 36 = 216, and 12 \times 18 = 216. The result lets you find one number from the others:
- L.C.M. = product \div H.C.F.
- H.C.F. = product \div L.C.M.
- One number = (H.C.F. \times L.C.M.) \div the other number.
For example, if the H.C.F. is 6, the L.C.M. is 36 and one number is 12, the other is (6 \times 36) \div 12 = 18. This relation is the basis of Exercise 4(C) questions 3 to 5.
What to practise from factors and multiples class 6 ICSE
The exercises are the main source of practice for this factors and multiples class 6 ICSE chapter. They come from the same book that continues into Chapter 5: Fractions.
- EXERCISE 4(A) — 5 questions on factors, prime numbers and prime factors.
- EXERCISE 4(B) — 8 questions on H.C.F. by the common factor, prime factor and division methods, including co-prime pairs.
- EXERCISE 4(C) — 8 questions on L.C.M. by all three methods and the H.C.F. \times L.C.M. relation.
- Revision Exercise (Chapter 4) — 8 questions mixing H.C.F., L.C.M. and true-or-false statements.
Note how the exercises phrase remainder problems. Exercise 4(B) questions 6 to 8 and Exercise 4(C) questions 6 to 8 use phrases like “leaves no remainder”, “decreased by” and “increased by”. A number that leaves no remainder when divided by 12, 15 and 18 is their L.C.M. A number that, when increased by 3, divides exactly is the L.C.M. minus 3. Read those phrases carefully — they tell you whether to add or subtract.
The official CISCE website publishes the Class 6 syllabus that guides this book.
Factors and multiples class 6 ICSE frequently asked questions
What is the difference between factors and multiples in factors and multiples class 6 ICSE maths?
A factor divides a number completely, so the factor list is finite. A multiple is the product of the number with 1, 2, 3 and so on, so the list never ends. For example, the factors of 6 are 1, 2, 3 and 6, while the multiples of 6 are 6, 12, 18, 24 and so on.
Why is 1 not a prime number in factors and multiples class 6 ICSE?
A prime number must have exactly two factors: 1 and itself. The number 1 has only one factor, itself, so it fails the test. That is why 1 is neither prime nor composite.
How do I find the H.C.F. of three numbers using the division method in Class 6 ICSE Maths?
Find the H.C.F. of any two of the numbers first. Then find the H.C.F. of that result and the third number. For 21, 28 and 49, the H.C.F. of 21 and 28 is 7, and the H.C.F. of 7 and 49 is 7, so the answer is 7.
When should I use the common division method for L.C.M. in this ICSE chapter?
Use the common division method when the numbers are large or when listing multiples would take too long. It handles three or more numbers in one table. For small pairs, the common multiple method is quicker.
What is the relation between H.C.F. and L.C.M. in factors and multiples class 6 ICSE?
For any two numbers, H.C.F. \times L.C.M. equals the product of the two numbers. If you know any three of the four values, you can find the fourth. For example, if the product is 120 and the H.C.F. is 4, then the L.C.M. is 120 \div 4 = 30.
Reference: ICSE Class 6 Mathematics textbook material, chapter 4: Factors and Multiples — CISCE-aligned.
More from this book
Explore Class 6 Maths Notes
Related chapters:
- Number System notes
- Integers notes
- Decimal Fractions notes