Back

Factors & Remainders

ALL MODULES

CAT 2025 Lesson : Factors & Remainders - HCF & LCM

In this lesson we will look at the basics of HCF & LCM, factors & multiples and then cover remainder theorem in detail. Please read the previous lesson on Number Theory before you start this.

1. Factors and Multiples

For any natural number

n

, a factor of

\bm{n}

is a number that perfectly divides

n

.

For any natural number

n

, a multiple of $\bm{n}$ is a number that is perfectly divisible by

\bm{n}

.
In other words, when we multiply

n

with another natural number, we get a multiple of

n

.

For instance, for the number

12

2

is a factor as

\dfrac{12}{2} = 6

and

60

is a multiple as

12 \times 5 = 60

1.1 HCF and LCM

The Highest Common Factor (HCF) of two or more numbers is the largest number that perfectly divides both the numbers.

E.g.,

5

is the largest number that divides both

10

and

15

. ∴ HCF (

10

15

) =

5

.

The Least Common Multiple (LCM) of two or more numbers is the smallest number that is a multiple of each of the numbers.

E.g.,

30

is the smallest number that is a multiple of both

10

and

15

. ∴ LCM (

10

15

) =

30

.

Note that:
1) HCF of two or more numbers will always be equal to or less than the smallest number.
2) LCM of two or more numbers will always be greater than or equal to the largest number.
3) LCM $\geq$ HCF

1.2 Using Division Method to calculate HCF and LCM

1.2.1 Successive Division for HCF

Successive division method is used to find HCF for

2

numbers. To determine the HCF,

(1) The larger number (dividend) is divided by the smaller number (divisor).
(2) If the remainder is not

0

, then the divisor becomes the dividend and the remainder becomes the divisor for the next division.
(3) This process continues till the remainder becomes

0

. The final divisor is the HCF of the two numbers.

Example 1

What is the HCF of

432

and

204

Solution

The final divisor here is

12

, which is the HCF of

432

and

204

.

Answer:

12

As successive division can be used for only

2

numbers, to find HCF of

3

numbers

a

b

and

c

, we need to find HCF (

a

b

) and then find HCF(

c

, HCF(

a

b

)). This becomes cumbersome.

∴ When more than

2

numbers are involved, Common Division and Prime Factorisation methods are recommended.

1.2.2 Common Division Method for HCF

To determine the HCF,
(1) If a prime number perfectly divides all the given numbers, then write this as the divisor to the left and write the respective quotients below the given numbers.
(2) The quotient from a particular division becomes the dividend in the next division.
(3) Repeat this process till there is no other number that perfectly divides all the dividends.
(4) HCF is the product of all the divisors.

Example 2

What is the HCF of

24

36

and

60

Solution

HCF

= 2 \times 2 \times 3 = 12

Answer:

12

1.2.3 Common Division Method for LCM

To determine the LCM,
(1) Each of the numbers are to be divided by prime numbers starting from the smallest.
(2) If a number is divisible by the prime, write the quotient below,else write the number as it is without division.
(3) Repeat this process till all the dividends become

1

.
(4) LCM is the product of all the divisors.

Example 3

What is the LCM of

24

36

and

90

Solution

LCM

= 2 \times 2 \times 2 \times 3 \times 3\times 5 = 8 \times 9 \times 5 = 360

Answer:

360

1.3 Prime Factorisation

A prime factor of an integer

n

is a prime number that perfectly divides

n

. All integers can be represented as the product of their prime factors.
E.g.,

4 = 2^2

;

37 = 37^1

;

180 = 2^2 \times 3^2 \times 5^1

1.3.1 Prime Factorisation Method - HCF

To determine the HCF:
(1) Prime factorise each of the numbers.
(2) For each of the numbers, determine the prime factors with their lowest powers.
(3) HCF is the product of the prime factors with their respective lowest powers determined in step

2

Example 4

What is the HCF of

24

36

and

60

Solution

24 = 2^3 \times 3^1

36 = 2^2 \times 3^2

60 = 2^2 \times 3^1 \times 5^1

HCF

= 2^2 \times 3^1 \times 5^0 = 12

Answer:

12

1.3.2 Prime Factorisation Method - LCM

To determine the LCM:
(1) Prime factorise each of the numbers.
(2) For each of the numbers, determine the prime factors with their highest powers.
(3) LCM is the product of the prime factors with their respective highest powers determined in step

2

Example 5

What is the LCM of

24

36

and

60

Solution

24 = 2^3 \times 3^1

36 = 2^2 \times 3^2

60 = 2^2 \times 3^1 \times 5^1

LCM

= 2^3 \times 3^2 \times 5^1 = 360

Answer:

360

1.4 HCF and LCM of two numbers

For any two natural numbers

a

and

b

,

1)

a \times b =

HCF(

a

b

)

\times

LCM (

a

b

)

2) If HCF is one of the numbers, then LCM is the other number.

Example 6

If the product of two numbers is

72

and their HCF is

6

, then what is their LCM?

Solution

a \times b =

HCF(

a

b

)

\times

LCM (

a

b

)

⇒

72 = 6 \times

LCM (

a

b

)

⇒ LCM (

a

b

)

= 12

Answer:

12

1.5 HCF and LCM of Fractions

HCF of fractions is

\dfrac{\mathrm{HCF \ of \ numerators}}{\mathrm{LCM \ of \ denominators}}

LCM of fractions is

\dfrac{\mathrm{LCM \ of \ numerators}}{\mathrm{HCF \ of \ denominators}}

Example 7

Find the HCF and LCM of

\dfrac{2}{7}, \dfrac{3}{14}, \dfrac{9}{35}

Solution

HCF =

\dfrac{\mathrm{HCF}(2, 3, 9)}{\mathrm{LCM}(7,14,35)} = \dfrac{1}{70}

LCM =

\dfrac{\mathrm{LCM}(2, 3, 9)}{\mathrm{HCF}(7,14,35)} = \dfrac{18}{7}

Want to read the full content

Unlock this content & enjoy all the features of the platform

Subscribe Now