Today In this session we will see { HCF LCM } the Highest Common Factor And Least Common Multiple of numbers and polynomials with detailed examples, questions, calculator, and worksheet for students.

## Highest Common Factor And Least Common Multiple [ L.C.M. & H.C.F. ]

Let’s see the small terms used in the LCM and HCF calculation. These terms are **factor, common factor**.

### Factor

The denominator or factor of any given number or expression is the number or expression that exactly divides that number or expression.

Any small number that can divide the big number completely called the factor of that number.

In case of polynomial, factor is the smallest expression that divides polynomial completely.

Examples:

2, 4, 5, 10, are the factors of 20.

x, x^{2}, x^{3}, x^{4} expressions are the factor of x^{5}.

**Note: ***1 is the factor of all numbers and polynomials. Any number or polynomial is a factor of itself.*

### Common Factor

A common factor is that number or expression that can divide two or more than two numbers completely.

Example: 3 is the common factor of 15 and 18 numbers. Because it can divide both numbers completely.

In the same way^{,} x is the common factor of X^{4} and x^{8}.

**Pro Tips:** **Factor is also called a divisor.Common Factor is also called as Common Divisor.**

## Highest Common Factor H.C.F.

Highest common factor: The greatest number or polynomial which exactly divides each of the given two or more polynomials or numbers is called the greatest common factor of the given polynomials or numbers. HCF is the short form of this Highest Common Factor.

#### Examples: Find the HCF of 6, 12, and 18.

Factors of these numbers

6 = 2X3

12 = 2X2X3

18 = 2X3X3

Here we can see 2X3 are common. Hence HCF = 2X3 = 6.

In the case of polynomials, Highest Common Factor is that divisor that has the highest power and the product of the highest power polynomial is always positive.

### Greatest Common Divisor By Factorization Method

In this method First of all we factorize the polynomial and then find the highest common factor. Highest Common Factor or HCF of polynomials is the product of similar divisors of polynomials.

See the below Example:

Find HCF of (x^{3}-x) and 3(x-1).

Solution:

Do factorizations of polynomials: (x^{3}-x) and 3(x-1)

(x^{3}-x) = x(x^{2}-1)

=x(x-1)(x+1)

3(x-1) = 3(x-1)

Here common divisor is (x-1).

Hence the HCF is (x-1)

H.C.F. of Polynomials By Factorization Method Top 20 Questions And Answers

## Least Common Multiple ( LCM )

LCM: The smallest number that can be completely divisible from the two given numbers is called the LCM or the Least Common Multiple of these two numbers.

In case of polynomials, LCM is the smallest power polynomial that is completely divisible by the two or more then two polynomials, is called the LCM of these given polynomials.

While finding LCM we have to deal with Multiple and Common Multiple. Let’s see each of them seprately.

### Multiple:

Multiple is that number or polynomial, that could be completely divisible from the given number or polynomial.

Example: 3, 6, 9, 21, 24 all are completely divisible by 3. Hence 3 is the multiple of these numbers.

In the same way, x is the multiple of x2, x5, x7, etc.

### Common Multiple:

Multiple is that number or polynomial, that could be completely divisible from two or more then two given numbers or polynomials.

Example For Numbers:

- 4, 12 are the common multiple of 24, 36.
- Polynomials (x+2), (x+3) are the common multiple of the polynomial (x
^{2}+5x+6).

LCM of Polynomials By Factorization Method Top 20 Questions And Answers.

## Relationship Between LCM and HCF

For Numbers:

**Multiplications of Numbers = LCM x HCF**

For Polynomials:

**Multiplication of Polynomials = Multiplication of LCM and HCF**

**Polynomial 1 x Polynomial 2 = LCM x HCF**

## Highest common factor and lowest common multiple questions

Que.1 Find the LCM of x^{2} +x – 6 and x^{3} – 8

Solution: In this question first of all we will do the factors of these polynomials (x^{2} +x – 6) and (x^{3} – 8)

The factors of polynomial x^{2} +x – 6 by middle term split method

x^{2} +x – 6 = x2 +3x -2x -6

x^{2} +x – 6 = 2(x+3)-2(x+3)

x^{2} +x – 6 = (x-2)(x+3)

The factors of x^{3} – 8 by a3 – b3 formula

x^{3} – 8 = x^{3} – 2^{3}

x^{3} – 8 = (x-2)(x^{2} + 4 +2x)

x^{3} – 8 = (x-2)(x^{2} + 2x + 4)

Here we can see that (x-2) is the common divisor.

LCM of (x^{2} +x – 6) and (x^{3} – 8) polynomials is (x-2)(x+3)(x^{2} + 2x + 4).

**Question: Find LCM and HCF of 510 & 92 numbers and also verify the relations**

Solution:

First of all we will do the factors of these 510 and 92 numbers.

Factors of 510 are as follow:

510 = 2X3X5X17

factors of 92 are as follow:

92 = 2X2X23

From the factors we can see 2 is the highest common factor of these two numbers.

Hence HCF = 2.

Least Common Multiple of these two numbers is = 2X3X5X17X2X23 = 23460.

So, LCM of 510 and 92 = 23460.

Relationship Verification:

as per the standard formula:

Multiplication of numbers = multiplication of LCM and HCF

510 X 92 = 23460 X 2

46920 = 46920

Hence:

Left hand side = Right Hand Side

Relationship is verified.

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