## Algebra Chapter 8 Lowest Common Multiple

**Lowest Common Multiple Introduction**

In arithmetic, you have learned how to find the H.C.F. i.e., the Highest Common Factor, and the L.C.M. i.e., the Lowest Common Multiple of numbers. In a similar way, we can find the H.C.F. and L.C.M. of two or more algebraic expressions. In this chapter, our aim is to find the L.C.M. of algebraic expressions by the method of factorization.

**Multiple**

When an expression is divisible by another expression, then the first expression is called a multiple of the second expression. For example, x^{2}y^{2} is a multiple of the expressions, x, y, xy, x^{2}y, xy^{2}, etc.

**Common Multiple**

When an expression is divisible by each of two or more expressions, then the first expression is called the common multiple of those expressions. For example, xy, x^{2}y^{2}, xy^{2}, etc. are divisible by each of * x* and

*and hence they are common multiples of x and y.*

*y***The Lowest Common Multiple**

Among the common multiples of some quantities, the lowest one (which is of the lowest power) is called the Lowest Common Multiple or L.C.M. of those quantities.

**For Example** **:**

L.C.M. of ab, a^{2}b, and ab^{2} is a^{2}b^{2}. Here a^{2}b^{2} is divisible by each of the quantities ab, a^{2}b, and ab^{2}. Moreover, a^{2}b^{2} is of minimum power among other quantities which are divisible by ab, a^{2}b, and ab^{2 }(for example, a^{3}b^{2}, a^{2}b^{3}, a^{4}6^{4}, etc.).

**Determination of L.C.M. by factorization**

1. First of all, the given expressions are resolved into factors.

2. The L.C.M. is the product of all kinds of factors in their highest powers.

3. If numerical coefficients appear before the alphabetic expressions then after finding the L.C.M. of the alphabetic expressions the arithmetical L.C.M. of the numerical coefficients is written before it.

**Read And Learn More WBBSE Solutions For Class 8 Maths**

## Algebra Chapter 8 Lowest Common Multiple Some Examples of L.C.M.

**Example 1**

**Find the L.C.M. of x ^{2}yz, xy^{2}z, and xyz^{2}.**

**Solution :**

Given: x^{2}yz, xy^{2}z, And xyz^{2}

Here, the factor with the highest power of x is the x^{2}

The factor with the highest power of y is y^{2 }

The factor with the highest power of z is z^{2 }

Hence, the required

L.C.M. =x^{2}y^{2}z^{2}

**Example 2**

**Find the L.C.M. of 4a ^{2}6c^{2}, 8ab^{2}c^{3} and 16a^{4}b^{3}c.**

**Solution :**

Given: 4a^{2}6c^{2}, 8ab^{2}c^{3} And 16a^{4}b^{3}c.

The L.C.M. of 4, 8, and 16 is 16.

The factor with the highest power of a is a^{4 }

The factor with the highest power of b is b^{3 }

The factor with the highest power of c is c^{3}

Hence, the required L.C.M. = 16a^{4}b^{3}c^{3}

**Example 3**

**Find the L.C.M. of a ^{3}b – ab^{3} and a^{3}b^{2} + a^{2}b^{3}.**

**Solution:**

Given a^{3}b – ab^{3} And a^{3}b^{2} + a^{2}b^{3}

First expression = a^{3}b – ab^{3}

= ab(a^{2} – b^{2}) = ab(a +b) (a – b)

Second expression

= a^{3}6^{2} + a^{2}b^{3 }

= a^{2}b^{2}(a + b)

Hence, the required L.C.M.

= a^{2}b^{2}(a + b) (a – b)

= a^{2}b^{2}(a^{2} – b^{2})

The required L.C.M = a^{2}b^{2}(a^{2} – b^{2})

**Example 4**

**Find the L.C.M. of x ^{2} – 4x + 3 and x^{2} – 5x + 6.**

**Solution :**

Given x^{2} * –* 4x + 3 And x

^{2}

*5x + 6*

*–*First expression = x^{2} – 4x + 3

= x^{2}-3x-x + 3

= x(x – 3) – l(x – 3)

= (x – 3) (x – 1)

Second expression

= x^{2} – 5x + 6

= x^{2} – 3x – 2x + 6

= x(x – 3) – 2(x – 3)

= (x-3)(x-2)

Hence, the required L.C.M. = (x – 1) (x – 2) (x – 3)

**Example 5**

**Find the L.C.M. of 8(a ^{2}– 4), 12(a^{3} + 8) and 36(a^{2} – 3a – 10).**

**Solution :**

Given 8(a^{2}– 4), 12(a^{3} + 8) And 36(a^{2} – 3a – 10).

First expression

= 8(a^{2} – 4)

= 2^{3} x {(a)^{2} – (2)^{2}}

= 2^{3}x (a+2) (a- 2)

Second expression

= 12(a^{3} + 8)

= 2^{2} x 3 x {(a)^{3} + (2)^{3}}

= 2^{2} x 3 x (a + 2) (a^{2} – 2a + 4)

Third expression

= 36 (a^{2} – 3a – 10)

= 2^{2} x 3^{2} x {a^{2} – 5a + 2a – 10}

= 2^{2} x 3^{2} x {a(a – 5) + 2(a – 5)}

= 2^{2} x 3^{2} x (a – 5) (a + 2)

Hence, the required L.C.M.

= 2^{3} x 3^{2} x (a + 2) (a – 2) (a^{2}-2a + 4) (a – 5)

= 72 (a + 2) (a – 2) (a – 5) (a^{2} – 2a + 4)

The required L.C.M = 72 (a + 2) (a – 2) (a – 5) (a^{2} – 2a + 4)

**Example 6**

**Find the L.C.M. of x ^{3}– 1, x^{4}– 1,x^{4} + x^{2}+ 1.**

**Solution:**

Given x^{3}– 1, x^{4}– 1,x^{4} + x^{2}+ 1

First expression

= x^{3} – 1

= (x)^{3} – (1)^{3 }

= (x – 1) (x^{2} + x + 1)

Second expression

= x^{4} – 1

= (x^{2})^{2} – (l)^{2}

= (x^{2} + 1) (x^{2} – 1)

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

Third expression

= x^{4} + x^{2} + 1

= (x^{2})^{2} + 2.x^{2}.1+ (1)^{2} -x?

= (x^{2} + 1)^{2} – (x)^{2 }

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

= (x^{2} + x + 1) (x^{2} – x + 1)

Hence, the required L.C.M.

= (x – 1) (x^{2} + x +1) (x +1) (x^{2} + 1) (x^{2} – X + 1)

= (x – 1) (x + 1) (x^{2} + 1) (x^{2} + x + 1) (x^{2} – x + 1)

The required L.C.M = (x – 1) (x + 1) (x^{2} + 1) (x^{2} + x + 1) (x^{2} – x + 1)

**Example 7**

**Find the L.C.M. of x ^{2} – y^{2}, x^{3} – y^{3}, 3X^{2} – 5x.y+ 2y^{2}.**

**Solution :**

Given x^{2} – y^{2}, x^{3} – y^{3}, 3X^{2} – 5x.y+ 2y^{2}

First expression

= x^{2} – y^{2 }

= (x + y) (x – y)

Second expression

= x^{3} – y^{3 }

= (x – y) (x^{2} + xy +y^{2})

Third expression

= 3X^{2} – 5x^{2} + 2y^{2 }

= 3X^{2} – 3xy – 2x^{2} + 2y^{2 }

= 3x(x -y)~ 2y(x – y)

= (x-y) (3x – 2y)

Hence, the required L.C.M.

= (x+y) (x-y) (x^{2} + xy +y^{2}) (3x- 2y)

= (x+y) (x-y)(3x- 2y)(x^{2} + xy +y^{2})

The required L.C.M = (x+y) (x-y)(3x- 2y)(x^{2} + xy +y^{2})

**Example 8**

**Find the L.C.M. of x ^{2} – y^{2} – z^{2} + 2yz, (x + y – z)^{2} and x^{2} + z^{2} – y^{2} + 2xz.**

**Solution :**

Given x^{2} – y^{2} – z^{2} + 2yz, (x + y – z)^{2} And x^{2} + z^{2} – y^{2} + 2xz.

First expression = x^{2} – y^{2} – z^{2} + 2yz = x^{2} – (y^{2} – 2yz + z^{2})

= (x)^{2} -xy-z)^{2 }= (x + y-z) (x-y +z)

Second expression

= (x + y – z)^{2 }

Third expression

= x^{2} + z^{2} – y^{2} + 2xz

= x^{2} + 2xz + z^{2} -y^{2 }

= (x + z)^{2} – (y)^{2 }

= (x + z + y) (x + z – y)

= (x + y + z) (x – y + z)

Hence, the required L.C.M.

= (x + y- z)^{2} (x-y+z) (x+y+z)

= (x-y+z) (x+y+z) (x + y- z)^{2}

The required L.C.M = (x-y+z) (x+y+z) (x + y- z)^{2}

**Example 9**

**Find the L.C.M. of x ^{3} – 16x, 2x^{3} + 9x^{2} + 4x and x + 4.**

**Solution :**

Given x^{3} – 16x, 2x^{3} + 9x^{2} + 4x And x + 4.

First expression

= x^{3} – 16x

= xfx^{2} – 16)

= x{(x)^{2} – (4)^{2}}

= x(x + 4)(x – 4)

Second expression

= 2x^{3} + 9x^{2} + 4x

= x(2x^{2} + 9x + 4)

= x(2x^{2} + 8x + x + 4)

= x{2x(x + 4) + l(x + 4)}

= x(x + 4)(2x + 1)

Third expression

= x + 4

Hence, the required L.C.M.

= x(x + 4)(x – 4)(2x + 1)

The required L.C.M = x(x + 4)(x – 4)(2x + 1)

**Example 10**

**Find the L.C.M. of a ^{2} – 6^{2} + c^{2} + 2ac, a^{2} – 6^{2} – c^{2} + 26c and ab + ac + b^{2} – c^{2}.**

**Solution :**

Given a^{2} – 6^{2} + c^{2} + 2ac, a^{2} – 6^{2} – c^{2} + 26c And ab + ac + b^{2} – c^{2}.

First expression

= a^{2} – b^{2} + c^{2} + 2ac

= a^{2} + 2ac + c^{2} – b^{2}

= (a + c)^{2} – (b)^{2 }

= (a + c + b)(a + c – b)

= (a + b + c)(a – b + c)

Second expression

= a^{2} – b^{2} – c^{2} + 2bc

= a^{2} – (b^{2} – 2bc + c^{2})

= (a)^{2} – (b – c)^{2 }

= (a + b – c)(a – b + c)

Third expression

= ab + ac + b^{2} – c^{2 }

= a(b + c) + (b + c)(b – c)

= (b + c)(a + b – c)

Hence, the required L.C.M.

= (a + b + c)(a – b + c)(a + b – c)(b + c)

= (a + b + c)(a – b + c)(a + b – c)(b + c)

The required L.C.M = (a + b + c)(a – b + c)(a + b – c)(b + c)

**Example 11**

**Find the L.C.M. of x ^{2} – xy – 2y^{2}, 2x^{2} – 5xy + 2y^{2} and 2x^{2} + xy – y^{2}.**

**Solution :**

Given x^{2} – xy – 2y^{2}, 2x^{2} – 5xy + 2y^{2} And 2x^{2} + xy – y^{2}.

First expression

= x^{2} – xy – 2y^{2 }

= x^{2} – 2xy + xy – 2y^{2}

= x(x – 2y) + y(x – 2y)

= (x- 2y)(x +y)

Second expression

= 2X^{2} – 5xy + 2y^{2 }

= 2x^{2} – 4xy – xy + 2y^{2 }

= 2x(x – 2y) – y(x – 2y)

= (x-2y)(2x-y)

Third expression

= 2x^{2} + xy – y^{2 }

= 2x^{2} + 2xy – xy -y^{2}

= 2x(x + y) – y(x + y)

= (x + y)(2x – y)

Hence, the required L.C.M.

= (x – 2y)(x + y)(2x – y)

The required L.C.M = (x – 2y)(x + y)(2x – y)

**Example 12**

**Find the L.C.M. of 3(x ^{2} – 9), 9(x^{3} + 27) and 27(x2 – 3x + 9).**

**Solution :**

Given 3(x^{2} – 9), 9(x^{3} + 27) and 27(x2 – 3x + 9)

First expression

= 3(x^{2} – 9)

= 3{(x)^{2} – (3)^{2}}

= 3(x + 3)(x – 3)

Second expression

= 9(x^{3} + 27)

= 9{(x)^{3} + (3)^{3}}

= 3^{2} (x + 3)(x^{2} – 3x + 9)

Third expression

= 27(x^{2} – 3x + 9)

= 3^{3} (x^{2} – 3x + 9)

Hence, the required L.C.M.

= 3^{3} (x + 3)(x – 3)(x^{2} – 3x + 9)

= 27(x + 3)(x – 3)(x^{2} – 3x + 9)

The required L.C.M = 27(x + 3)(x – 3)(x^{2} – 3x + 9)