ICSE Class 6 Maths Chapter 04 Factors and Multiples

Chapter 4: Factors And Multiples

(Including H.C.F. and L.C.M.)

Factors

When two or more natural numbers are multiplied together, the result is referred to as their product, and each of the numbers multiplied is called a factor of this product.

For example:

(i) Product of 5 and 7 = 5 × 7 = 35; therefore 5 and 7 are factors of 35.

(ii) Product of 2, 3 and 7 = 2 × 3 × 7 = 42; therefore each of 2, 3 and 7 is a factor of 42.

In other words:

Any natural number that divides a given natural number completely is called a factor of the given number.

For example:

(i) 5 divides 20 completely → 5 is a factor of 20

(ii) 6 divides 12 completely → 6 is a factor of 12

(iii) 15 divides 30 completely → 15 is a factor of 30 and so on.

Now consider the following examples:

(i) 24 = 1 × 24 → 1 and 24 are factors of 24.

(ii) 24 = 2 × 12 → 2 and 12 are factors of 24.

(iii) 24 = 3 × 8 → 3 and 8 are factors of 24.

(iv) 24 = 4 × 6 → 4 and 6 are factors of 24.

Combining, we get:

1, 2, 3, 4, 6, 8, 12 and 24 are factors of 24.

∴ Factors of 24 = F₂₄ = 1, 2, 3, 4, 6, 8, 12 and 24

Each of 1, 2, 3, 4, 6, 8, 12 and 24 divides 24 completely.

In the same way,

(i) F₃₀ = Factors of 30 = 1, 2, 3, 5, 6, 10, 15 and 30

Factors of 6 = Each natural number that divides 6 completely = 1, 2, 3 and 6

(ii) F₁₈ = 1, 2, 3, 6, 9 and 18

(iii) F₄₅ = 1, 3, 5, 9, 15 and 45 and so on.

1 (one) is a factor of every number.

Every number is a factor of itself.

Teacher's Note

Understanding factors helps in dividing objects into equal groups, such as arranging 24 students into equal rows for a classroom activity.

Example 1

Write the factors of (i) 13 (ii) 25 (iii) 28

Solution

(i) Factors of 13 = F₁₃ = 1 and 13 (Ans.)

(ii) F₂₅ = 1, 5 and 25 (Ans.)

(iii) F₂₈ = 1, 2, 4, 7, 14 and 28 (Ans.)

Prime Numbers

A natural number that is divisible only by 1 (one) and itself is called a prime number.

For example:

(i) 2 is divisible only by 1 (one) and itself; therefore 2 is a prime number.

(ii) 3 is divisible only by 1 and itself; therefore 3 is a prime number.

(iii) 4 is divisible by 1 and itself as well as by 2; so 4 is not a prime number.

In the same way; we find:

(iv) each of 5, 7, 11, 13, 17, ... is a prime number.

(v) none of 1, 4, 6, 8, 9, 10, 12, ... is a prime number.

1. Every prime number is greater than 1 (one).

2. Two (2) is the smallest prime number.

3. All prime numbers except 2 are odd.

To be more clear, note:

If a natural number has only two factors, it is a prime number.

→ (i) 1 (one) is not a prime number as it has only one factor, 1, which is itself.

(ii) 7 is a prime number as it has only two factors, 1 and 7.

(iii) 10 is not a prime number as it has more than two factors: 1, 2, 5 and 10.

Every natural number that has more than two factors is called a composite number.

Since, factors of 10 are 1, 2, 5 and 10

∴ 10 is a composite number

Similarly, each of 4, 6, 8, 9, 10, 12, 14, ..., 24, 25, 26, 27, etc. is a composite number.

Every even number greater than 2 is a composite number.

Teacher's Note

Prime numbers are used in cryptography and security systems, making them essential for protecting data in modern digital communication.

Prime Numbers From 1 To 100

Eratosthenes, a Greek scholar, used the following method to distinguish the prime numbers from among the natural numbers. For this reason, this method is known as the Sieve of Eratosthenes.

Step 1: Write the natural numbers 1 to 100 in rows of 10 as shown below:

1 (2) (3) 4 (5) 6 (7) 8 9 10

(11) 12 (13) 14 15 16 (17) 18 (19) 20

21 22 (23) 24 25 26 27 28 (29) 30

(31) 32 33 34 35 36 (37) 38 39 40

(41) 42 (43) 44 45 46 (47) 48 49 50

51 52 (53) 54 55 56 57 58 (59) 60

(61) 62 63 64 65 66 (67) 68 69 70

(71) 72 (73) 74 75 76 77 78 (79) 80

81 82 (83) 84 85 86 87 88 (89) 90

91 92 93 94 95 96 (97) 98 99 100

Step 2: 1 (one) is not a prime number; strike it out.

Step 3: The first, i.e. the smallest, prime number is 2; encircle 2 and strike out all other numbers which are divisible by 2.

Step 4: The next prime number is 3; encircle 3 and strike out all other numbers that are divisible by 3.

(Some of these numbers are already cut as they are divisible by 2 as well.)

Step 5: The next prime number is 5; encircle 5 and strike out all other numbers that are divisible by 5.

(Some of these numbers are already cut as they are divisible by 2 and/or 3.)

Step 6: Adopt the same steps for prime numbers 7, 11, 13 and so on.

Finally, some numbers emerge encircled, and they are the prime numbers between 1 and 100.

Prime Factors

Let us take a number, say, 24. The factors of 24 are: 1, 2, 3, 4, 6, 8, 12 and 24. Out of these factors, 2 and 3 are prime numbers, and so the factors 2 and 3 are called the prime factors of 24.

Thus, factors of 24 = 1, 2, 3, 4, 6, 8, 12 and 24

→ Prime factors of 24 = 2 and 3

In other words, we write:

F₂₄ = 1, 2, 3, 4, 6, 8, 12 and 24; and P.F.₂₄ = 2 and 3

In the same way:

(i) F₅₀ = 1, 2, 5, 10, 25 and 50 → P.F.₅₀ = 2 and 5

(ii) F₆₄ = 1, 2, 4, 8, 16, 32 and 64 → P.F.₆₄ = 2 and so on.

Teacher's Note

Prime factorization helps break down numbers into their simplest building blocks, similar to understanding how complex machines are built from basic components.

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