By studying this lesson, you will be able to

• find the factors of a whole number,

• write the multiples of a whole number,

• write the prime factors of a whole number,

• find the highest common factor (HCF) of a whole number and

• find the least common multiple (LCM) of a whole number.

4.2 Factors and multiples of a whole number

You learnt in grade six, how to find the factors and multiples of a whole number. Let us recall what you learnt.

Let us find the factors of 36.

Let us factorize 36 by expressing it as a product of two whole numbers.

Therefore, the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36.

Let us factorize 126, using the method of division.

Since the number 126, can be divided by 2 without remainder, 2 is a factor of 126.

Since, 2 × 63 = 126, we obtain that 63 is also a factor of 126.

2 × 63 = 126 

3 × 42 = 126 

6 × 21 = 126

7 × 18 = 126 

9 × 14 = 126 

14 × 9 = 126

Therefore, the factors of 126 are 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63 and 126.

Note:

The divisibility rules can be used to determine whether a given number is divisible by another number or not.

Now let us consider how multiples of a whole number are found.

Let us compute the multiples of 13.

This can be done by multiplying 13 by whole numbers.

13 × 1 = 13 

13 × 2 = 26 

13 × 3 = 39 

13 × 4 = 52

13, 26, 39 and 52 are a few examples of multiples of 13. Note that 13 is a factor of each of them. Therefore, any number of which 13 is a factor, is a multiple of 13.

(1) Factorize.

(i) 150 (ii) 204 (iii) 165 (iv) 284

(2) Write down the ten factors of 770 below 100.

(3) 

(i) Write five multiples of 36.

(ii) Write five multiples of 112.

(iii) Write five multiples of 53 below 500.

(4) 180 chairs in an examination hall have to be arranged such that each row has an equal number of chairs. If the minimum number of chairs that should be in a row is 10 and the maximum that could be in a row is 15, find how many possible ways there are to arrange the chairs.

4.3 Prime factors of a whole number

You have already learnt that whole numbers greater than one with exactly two distinct factors are called prime numbers.

Let us recall the prime numbers below 20.

They are 2, 3, 5, 7, 11, 13, 17 and 19.

Let us identify the prime factors of 36. We learnt above that the factors 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36.

There are only two prime numbers among them, namely, 2 and 3. These are the prime factors of 36.

Let us find the prime factors of 60.

The factors of 60 are 1, 2, 3, 4, 5, 6, 12, 15, 20, 30 and 60.

The prime factors among them are 2, 3 and 5.

The prime numbers among the factors of a number are its prime factors.

Any whole number which is not a prime number can be expressed as a product of its prime factors.

A method of finding the prime factors of a whole number using the method of division and writing the number as a product of its prime factors is described below.

Let us find the prime factors of 84 and write it as a product of its prime factors.

Let us write 75 as a product of its prime factors. Let us divide 75 by prime numbers.

• When finding the prime factors of a whole number, the number is divided by the prime numbers which divide it without remainder, starting from the smallest such prime number, until the answer 1 is obtained.

• The prime numbers which divide a number without remainder are its prime factors.

• To write a number as a product of its prime factors, write it as a product of the prime numbers by which it was divided.

(1) Find the prime factors of each of the following numbers.

(i) 81 (ii) 84 (iii) 96

(2) Express each of the following numbers as a product of its prime factors.

(i) 12 (ii) 15 (iii) 16 (iv) 18 (v) 20

(vi) 28 (vii) 59 (viii) 65 (ix) 77 (x) 91

4.4 Finding the factors of a number by considering its prime factors

Suppose we need to find the factors of 72.

Let us start by writing 72 as a product of its prime factors.

The factors of a whole number (which are not its prime factors or 1) can be obtained by taking products of 2 or more of its prime factors.

2, 36, 4, 18, 8, 9, 24 and 3 are eight factors of 72. 1 and 72 are also factors of 72.

1, 2, 3, 4, 8, 9, 18, 24, 36 and 72 are ten factors of 72.

(1) Find six factors of each of the following numbers by considering their prime factors.

(i) 20 (ii) 42 (iii) 70 (iv) 84 (v) 66 (vi) 99

4.5 Highest Common Factor (HCF) (Greatest Common Divisor (GCD))

Let us now consider what the highest common factor (HCF) of several numbers is and how it is found.

Let us find the highest common factor of the numbers 6, 12 and 18.

👉Write down the factors of these numbers as follows.

👉Circle and write the factors common to all three numbers.

The factors which are common to 6, 12 and 18 are, 1, 2, 3 and 6.

👉The largest number among the common factors is the highest common factor of these numbers.

We observe that the largest or the greatest of these common factors is 6.

Therefore, 6 is the highest common factor of 6, 12 and 18.

Thus, the highest common factor of 6, 12 and 18 is 6, which is the largest number by which these three numbers are divisible.

• Given two or more numbers, the largest number which is a factor of all of them is known as their highest common factor (HCF).

• Accordingly, the highest common factor is the largest number by which all the given numbers are divisible.

• If 1 is the only common divisor of several numbers, then the highest common factor of these numbers is 1.

• Finding the highest common factor by writing each number as a product of its prime factors

Let us find the highest common factor of 6, 12 and 18.

👉Let us write each number as a product of its prime factors.

👉The highest common factor is obtained by taking the product of the prime factors which are common to all three numbers.

2 and 3 are the common prime factors of 6, 12 and 18.

Thus, the HCF of 6, 12 and 18 is 2 × 3 = 6.

• Finding the highest common factor by the method of division

👉The result is 3, 6 and 9. Since 3, 6 and 9 are divisible by 3, the next smallest prime number, divide them by 3 and write the result below the respective numbers.

👉The result is 1, 2 and 3. Since there isn’t a prime number which divides all of 1, 2 and 3 without remainder, the division is stopped here.

👉The HCF is obtained by multiplying the numbers by which division was done.

Thus the HCF of 6, 12 and 18 is 2 × 3 = 6.

When using the method of division to find the HCF,

👉keep dividing all the numbers by the prime numbers which divide all the numbers without remainder.

👉then multiply all the divisors and obtain the HCF of the given numbers.

The HCF of any set of prime numbers is 1.

The HCF 36 of 72 and 108 can also be described as the largest number that divides them both without remainder.

(1) Fill in the blanks to obtain the HCF by writing down all factors of the given numbers.

(i) Factors of 8 are ...., ...., .... and .....

Factors of 12 are ...., ...., ...., ...., ...., and .....

Factors common to 8 and 12 are ...., ...., and .....

∴ The HCF of 8 and 12 is .... .

(ii) 54 written as a product of its prime factors = 2 × ''''' × 3 × '''''

90 written as a product of its prime factors = ''''' × 3 × ''''' × 5'

72 written as a product of its prime factors = 2 × 2 × ''''' × ''''' × '''''

∴ The HCF of 54, 90 and 72 = ''''' × ''''' × '''''

= '''''

(2) Find the HCF of each pair of numbers by writing down all their factors.

(i) 12, 15 (ii) 24, 30 (iii) 60, 72

(iv) 4, 5 (v) 72, 96 (vi) 54, 35

(3) Find the HCF of each pair of numbers by writing each number as a product of its prime factors.

(i) 24, 36 (ii) 45, 54 (iii) 32,48 (iv) 48, 72 (v) 18, 36

(4) Find the HCF by any method you like.

(i) 18, 12, 15 (ii) 12, 18, 24 (iii) 24, 32, 48 (iv) 18, 27, 36

4.6 Least Common Multiple (LCM)

Now let us consider what is meant by the least common multiple of several numbers and how it is found.

As an example, let us find the least common multiple of the numbers 2, 3 and 4.

👉List the multiples of the given numbers.

Several multiples of the numbers 2, 3 and 4 are given in the following table.

👉Circle and write down the common multiples. You will observe that the common multiples of the three numbers listed here are 12 and 24.

Further, if we continue to write the common multiples of 2, 3 and 4, we will obtain 12, 24, 36, 48, 60 etc

👉The smallest of the common multiples of several numbers is called the least common multiple (LCM) of these numbers.

The smallest or the least of the common multiples 12, 24, 36, 48, 60, ... of the numbers 2, 3 and 4 is 12.

Therefore, 12 is the least common multiple of 2, 3 and 4.

In other words, the smallest number which is divisible by 2, 3 and 4 is

the least common multiple of 2, 3 and 4.

The least common multiple of several numbers is the smallest positive number which is divisible by all these numbers.

Note

• The HCF of several numbers is equal to or smaller than the smallest of these numbers

• The LCM of several numbers is equal to or larger than the largest of these numbers.

• The HCF of any two numbers is smaller than the LCM of the two numbers.

• Finding the LCM of several numbers by considering their prime factors

Let us see how the LCM of several numbers is found by considering their prime factors.

Let us find the LCM of 4, 12 and 18.

👉Let us write each number as a product of its prime factors.

Let us select the greatest power of each prime factor.

There are two distinct prime factors, namely, 2 and 3. When the factors of all three numbers are considered,

👉The LCM of the given numbers is the product of these greatest powers.

• Finding the LCM by the method of division

• Finding the LCM by the method of division

Let us find the LCM of 4, 12 and 18.

👉We get 2, 6 and 9 as the result. No prime number divides all of them without remainder. However,

2 divides both 2 and 6 without remainder. Divide

2 and 6 by 2, and write the results below the respective numbers. Write 9 below 9.

👉Since 3 and 9 are divisible by 3, the next smallest prime number, divide them by 3 and write the results below the respective numbers. Now observe that we cannot find at least two numbers which are divisible by the same number. Therefore, the division is stopped here.

👉Multiply all divisors and all numbers left in the last row. The product gives the LCM of the given numbers.

Accordingly, the LCM of 4, 12 and 18 = 2 × 2 × 3 × 1 × 1 × 3 = 36

Note

When using the method of division to find the LCM, keep dividing if there remain at least two numbers, divisible by another and obtain the LCM of the given numbers as above.

Let us find the LCM of 4, 3 and 5.

Here, we do not have at least two numbers which are divisible by a common number which is greater than 1.

Therefore, the LCM of 4,3 and 5 = 4 × 3 × 5 = 60

(1) Find the LCM of each of the following triples of numbers.

(i) 18, 24, 36 (ii) 8, 14, 28 (iii) 20, 30, 40

(iv) 9, 12, 27 (v) 2, 3, 5 (vi) 36, 54, 24

82) At a military function, three cannons are fired at intervals of 12 seconds. 16 seconds and 18 seconds respectively. If the three cannons are fired together initially, after how many seconds will they all be fired together again?

(1) Without dividing, determine whether the number 35 343 is divisible by 3, 4, 6 and 9.

(2) Fill in the blanks.

(i) The HCF of 2 and 3 is .... .

(ii) The LCM of 4 and 12 is .... .

(iii) The HCF of two prime numbers is .... .

(iv) The LCM of 2, 3 and 5 is .... .

(3) Find the HCF and LCM of 12, 42 and 75.

(4) It is proposed to distribute books among 45 students in a class, such that each student receives no less than 5 books and no more than 10 books. Find all possible values that the total number of books that need to be bought can take, if all the students are to receive the same number of books and no books should be left over.