         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         