2.5 HCF(GCD) and LCM of Algebraic terms:

 

In the case of numbers, we have understood what HCF (GCD) of two or more numbers is. It is the highest among all common factors of the numbers.

 

Let us take an example of 4 numbers 4, 8, 20, 16. We know by inspection that 2 and 4 are common factors of all the four numbers and highest among them is 4.

Therefore HCF of 4, 8, 20, and 16 is 4.

 

HCF is useful in simplifying fractions.

 

Let us examine how we simplify the fraction 30/48.

First, we find HCF of 30 and 48 which happens to be 6.Then we express both numerator and denominator as a product of this HCF.

(30 = 6*5 and 48 = 6*8)

Next, we cancel this common factor from both numerator and denominator to get the proper fraction

(I.e. 30/48 = 6*5/6*8 = 5/8)

 

What is LCM?  It is the lowest number among all the common multiples of the numbers.

Let us take the same example of numbers 4, 8, 20, 16.

We notice that 80, 160, 320 … are all common multiples of the above numbers. The least among these common multiples is 80 and 80 is called LCM

LCM is useful in adding fractions.

 

Let us add 1/4, 1/8, 1/20

LCM of 4,8,20 is 40

1/4 = 10/40

1/8 = 5/40

1/20 = 2/40

1/4+1/8+1/20 = 10/40+5/40+2/40 = (10+5+2)/40 = 17/40

For finding HCF and LCM of algebraic terms, we follow same method that we use in the case of finding HCF  and LCM of numbers.

Let us recollect Finding HCF and LCM of numbers.

Finding HCF

Step	Procedure
1	List all the numbers
2	On the left side write the smallest common divisor for all  of these numbers
3	On the 2nd line write the quotients
4	On the left side write the common divisor for all  of these quotients as got in the step2
5	Repeat these steps till there are no more common factors for all the numbers

 

Product of the common divisors (appearing on the left hand side) is HCF of the numbers

Find HCF of  16,24,20

 

2 | 16,24,20

2 |  8,12,10

      4, 6, 5

 

We stop further divisions as the remaining numbers(all) ( 4 6 and 5) do not have common factors other than 1

 

Therefore (2*2)  = 4 is HCF of 16,24,20

 

Finding LCM

Step	Procedure
1	List all the numbers
2	On the left side write the common divisor for any of these numbers
3	On the 2nd line write the quotients
4	On the left side write the common divisor for any of these quotients as got in the step2
5	Repeat these steps till there are no more common factors among any two numbers

Product of the common divisors and the remaining numbers on the last line is LCM

 

Find LCM of  16,24,20

 

2 | 16,24,20

2 |  8,12,10

2 |  4,6,5

   |  2,3,5

 

We stop here further divisions as none of the remaining numbers (2,3,5)  have common factors other than 1

Thus LCM = (2*2*2)*(2*3*5) = 240

 

 

Also observe that,  product of HCF and  LCM  of 2 numbers = Product of  2 numbers.

 

This relationship holds good for algebraic terms also, hence if two terms and their HCF or LCM are given we can find their LCM or HCF respectively.

 

 

2.5 Problem 1 : Find HCF of  16a⁴b³x³, 24b²m³n⁴y, 20a²b³nx³

 

Solution:

1.HCF of  co-efficients (16,24,20) is 4

The variables are a⁴b³x³, b²m³n⁴y, a²b³nx³ and we notice that b is a common factor of the variables

 

4b | 16a⁴b³x³, 24b²m³n⁴y, 20a²b³nx³ ( We start dividing by 4b as it is the  common factor for all the terms) 

 b  |  4a⁴b²x³, 6bm³n⁴y, 5a²b²nx³ (b is a common factor  of  all terms)

     |4a⁴bx³, 6m³n⁴y, 5a²bnx³

 We need to stop further division as there are no more common factors among all the terms

4b*b= 4b²  is HCF of the term

HCF is useful for taking common factors out in addition/subtractions and in simplifying expressions

Let us simplify 16a⁴b³x³+24b²m³n⁴y- 20a²b³nx³

 

16a⁴b³x³+24b²m³n⁴y- 20a²b³nx³

=4b²(4a⁴bx³+6m³n⁴y- 5a²bnx³)

 

2.5 Problem 2 : Find HCF and LCM  of  6x²y³, 8x³y², 12x⁴y³, 10x³y⁴

 

Solution:

1.HCF of  co-efficients (6,8,12,10) is 2

The variables are x²y³, x³y², x⁴y³, x³y⁴and we notice that x is a common factor of the variables

a)   Finding HCF

 

2x | 6x²y³, 8x³y², 12x⁴y³, 10x³y⁴ (2x  is a common factor of all the terms)

x   | 3xy³, 4x²y², 6x³y³, 5x²y⁴ 

y   |3y³,   4xy²,   6x²y³, 5xy⁴

y   |3y²,   4xy,    6x²y², 5xy³

      3y,    4x,     6x²y,  5xy²

We need to stop further division as there are no more common factors among all the terms

2x*x*y*y = 2x²y² is HCF of variable

Use of HCF:

Let us simplify 6x²y³+8x³y²-12x⁴y³+10x³y⁴

 

6x²y³+8x³y²-12x⁴y³+10x³y⁴

= 2x²y²(3y+4x-6x²y+5xy²)

 

a)   Finding LCM

 

2x | 6x²y³, 8x³y², 12x⁴y³, 10x³y⁴ (2x  is a common factor of all the terms)

x   | 3xy³, 4x²y², 6x³y³, 5x²y⁴

y   |3y³,   4xy²,   6x²y³, 5xy⁴

y   |3y²,   4xy,    6x²y², 5xy³

Y  |3y,     4x,    6x²y   , 5xy²

x   | 3,      4x     6x²,     5xy

2  | 3,      4      6x       5y

_{­­­­­­­­­­­­­­} 3  | 3,      2      3x       5y

    | 1,     2      x        5y

 

We stop further division, as there are no more common factors among any two terms.

Thus LCM =  ( 2x*x*y*Y)*(Y*x*2*3*2*x*5y) =2x²y²* 60x²y² = 120x⁴y⁴

     

Use of LCM:

Simplify (1/6x²y³)+(1/8x³y²)-(1/12x⁴y³ )+(1/10x³y⁴)

Note

(1/6x²y³)  = (20x²y/120x⁴y⁴)

(1/8x³y²)  = (15xy²/120x⁴y⁴)

(1/12x⁴y³) = (10y/120x⁴y⁴)

(1/10x³y⁴) = (12x/120x⁴y⁴)

(1/6x²y³)+(1/8x³y²)-(1/12x⁴y³ )+(1/10x³y⁴)

= (20x²y+15xy²-10y+12x)÷(120x⁴y⁴)

 

 

 

 

2.5 Summary of learning

 

 

No	Points studied
1	Finding HCF and LCM of algebraic terms by division method