Subject: Compulsory Maths

The highest number that divides exactly into two or more numbers is called highest common factor(HCF) and lowest Common multiples are multiples that two numbers have in common. These can be useful when working with fractions and ratios.

The highest common factor (HCF) of the algebraic expression is the largest number that divides evenly into both numbers. It can be said as largest of all common factors.

For example, HCF of 6x^{3}y^{2} and 10x^{5}y^{4} is 2x^{3}y^{2} since

HCF of 6 and 10 is 2

HCF of x^{3} and x^{5} is x^{3}

and HCF of y^{2} and y^{4} is y^{2}

To find the HCF of compound expressions, first of all, resolve each expression into factors and then find HCF.

**Example:**

Find the HCF of 3x^{2}- 6x and x^{2}+ x - 6

Solution:

1st expression = 3x^{2}- 6x

= 3x(x - 2)

2nd expression = x^{2}+ x - 6

= x^{2}+ 3x - 2x - 6

= x(x + 3) - 2(x + 3)

= (x + 3)(x - 2)

∴ = x - 2

The lowest common multiple(LCM) is found by multiplying all the factors which appear on either list. LCM of any number is the smallest whole number which is multiple of both.

For example, LCM of 6x^{3}y^{2 }and 10x^{5}y^{4 }is 30 x^{5}y^{4 }since

LCM of 6 and 10 is 30, LCM of x^{3 }and x^{5 }and LCM of y^{2 }and y^{4 }is y^{4}.

To find the LCM of compound expressions, proceed as in the case of HCF and then find LCM.

**Example**

Find the LCM of 3x^{2}- 6x

1^{st} expression = 3x^{2}- 6x

= 3x(x - 2)

2^{nd} expression = x^{2}+ x - 6

= x^{2}+ 3x - 2x - 6

= x(x + 3) - 2(x + 3)

= (x + 3)(x - 2)

LCM = 3x(x - 2)(x + 3)

- H.C.F is the largest number that divides every into both numbers.
- H.C.F is useful when simplifying fraction.
- L.C.M is the smallest number that is a common multiple of two or more numbers.

- It includes every relationship which established among the people.
- There can be more than one community in a society. Community smaller than society.
- It is a network of social relationships which cannot see or touched.
- common interests and common objectives are not necessary for society.

Find the HCF of:

4x^{2}y and xy^{2}

Solution:

4x^{2}y and xy^{2}

Here, first expression = 4x^{2}y = 4 × x × x × y

The second expression = xy^{2}= x × y × y

Taking common of both expression = xy.

∴ H.C.F. = xy

Find the HCF of:

9x^{2}y^{3} and 15xy^{2}

Solution:

Here first expression = 9x^{2}y^{3}= 3 × 3 × x × x × y × y × y

Second expression = 15xy^{2}= 3 × 5 × x × y × y

Taking common from both expression

= 3 × x × y × y

∴ H.C.F. = 3xy^{2}

Find the HCF of:

a^{2}bc, b^{2} ac and c^{2}ab

Solution:

Here, first expression = a^{2}bc = a × a × b × c

Second expression=b^{2}ac= b × b × a × c

Third expression= b^{2}ac = b × b ×a× c

Taking common of the three expression=a × b × c

∴ H.C.F = abc

Find the HCF of x^{2}-4 and 3x+6

Solution:

Here given x^{2}-4 and 3x+6

First expression = x^{2}-4 = x^{2}-2^{2}= (x-2)(x+2)

Second expression = 3x+6 = 3(x+2)

∴ H.C.F = x+2

Find the HCF of x^{2}-y^{2} and xy - y^{2}

Solution:

Given, x^{2}-y^{2} and xy - y^{2}

First expression = x^{2}-y^{2}= (x+y) (x-y)

Second expression = xy - y^{2}= y(x-y)

Taking common from both expression = x-y

∴ H.C.F = x-y

Find the HCF of 3x^{2}-6x and x^{2}+x-6

Solution:

Here given,3x^{2}-6x and x^{2}+x-6

1st expression= 3x^{2}-6x

= 3x(x-2)

2nd expression= x^{2}+x-6

=x^{2}+3x-2x-6

=x(x+3)-2(x+3)

=(x+3)(x-2)

∴H.C.F = x-2

Find the HCF of 3a+b and 15a +5 b

Solution:

Here given,3a+b and 15a +5 b

1st expression=3a+b

2nd expression=15a+5b=5(3a+b)

Taking common from both expression =3a+b

∴H.C.F= 3a+b

Find the HCF of a^{3}+ 6a^{2}- 4 a-24; a^{2}+5a+6 and a^{2}-4

Solution:

Here given,a^{3}+ 6a^{2}- 4 a-24; a^{2}+5a+6 and a^{2}-4

1st expression=a^{3}+ 6a^{2}- 4 a-24

=a^{3}-4a+6a^{2}-24

=a(a^{2}-4)+6(a^{2}-4)

=(a^{2}-4)(a+6)

=(a-2)(a+2)(a+6)

2nd expression= a^{2}+5a+6

=a^{2}+(2+3)a+6

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

=a(a+2)+3(a+2)

=(a+2)(a+3)

3rd expression=a^{2}-4

=(a-2)(a+2)

Taking common from three expression=a+2

∴H.C.F= a+2

Find the HCF of (x-a), x^{2}-a^{2} and x^{2}-2ax+a^{2}

Solution:

Here given,(x-a), x^{2}-a^{2} and x^{2}-2ax+a^{2}

1st expression = x-a

2nd expression = x^{2}-a^{2}= (x-a)(x+a)

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

Taking common from three expression = x-a

∴ H.C.F = x-a

Find the HCF of a^{2}-3a+2; 3a^{2}-2a-8 and 2a^{2}-9a+10

Solution:

Here given, a^{2}-3a+2; 3a^{2}-2a-8 and 2a^{2}-9a+10

1st expression = a^{2}-3a+2

= a^{2}- 3a + 2

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

= a^{2}-a-2a+2

= a(a-1)-2(a-1)

= (a-1)(a-2)

2nd expression=3a^{2}-2a-8

= 3a^{2}-(6-4)a-8

= 3a^{2}-6a + 4a-8

= 3a(a-2)+4(a-2)

= (a-2)(3a+4)

3rd expression = 2a^{2}-9a+10

= 2a^{2}-(4+5)a+10

= 2a^{2}-4a-5a+10

= 2a(a-2)-5(a-2)

= (a-2)(2a-5)

Taking common of three expression = a+2

∴ HCF = a-2

Find the LCM of 3x^{2}-6x and x^{2}+x-6

Solution:

Here given, 3x^{2}-6x and x^{2}+x-6

1st expression = 3x^{2}-6x

= 3x(x-2)

2nd expression = x^{2 }+ x-6

= x^{2}+ 3x - 2x-6

= x(x+3)-2(x+3)

= (x+3)(x-2)

\(\therefore\) LCM = 3x(x-2)(x+3)

Find the LCM of:

2x and 4

Solution:

Here given, 2x and 4

1st expression = 2x = 2 × x

2nd expression = 4 = 2×2

LCM= 2×2× x = 4x

Find the H.C.F. and L.C.M. of the expressions a^{2} – 12a + 35 and a^{2} – 8a + 7 by factorization.

Solution:

First expression = a^{2} – 12a + 35

= a^{2} – 7a – 5a + 35

= a(a – 7) – 5(a – 7)

= (a – 7) (a – 5)

Second expression = a^{2} – 8a + 7

= a^{2} – 7a – a + 7

= a(a – 7) – 1(a – 7)

= (a – 7) (a – 1)

Therefore, the H.C.F. = (a – 7) and L.C.M. = (a – 7) (a – 5) (a – 1)

Find the L.C.M. of the two expressions a^{2} + 7a – 18, a^{2} + 10a + 9 with the help of their H.C.F.

Solution:

First expression = a^{2} + 7a – 18

= a^{2} + 9a – 2a – 18

= a(a + 9) – 2(a + 9)

= (a + 9) (a – 2)

Second expression = a^{2} + 10a + 9

= a^{2} + 9a + a + 9

= a(a + 9) + 1(a + 9)

= (a + 9) (a + 1)

Therefore, the H.C.F. = (a + 9)

Therefore, L.C.M. = Product of the two expressions H.C.F.

=\(\frac{(a^2+7a-18)(a^2+10a+9)}{(a+9)}\)

=\(\frac{(a+9)(a-2)(a+9)(a+1)}{(a+9)}\)

= (a – 2) (a + 9) (a + 1)

Find the L.C.M of m^{3} – 3m^{2} + 2m and m^{3} + m^{2} – 6m by factorization.

Solution:

First expression = m^{3} – 3m^{2} + 2m

= m(m^{2} – 3m + 2), by taking common ‘m’

= m(m^{2} - 2m - m + 2), by splitting the middle term -3m = -2m - m

= m[m(m - 2) - 1(m - 2)]

= m(m - 2) (m - 1)

= m × (m - 2) × (m - 1)

Second expression = m^{3} + m^{2} – 6m

= m(m^{2} + m - 6) by taking common ‘m’

= m(m^{2} + 3m – 2m - 6), by splitting the middle term m = 3m - 2m

= m[m(m + 3) - 2(m + 3)]

= m(m + 3)(m - 2)

= m × (m + 3) × (m - 2)

In both the expressions, the common factors are ‘m’ and ‘(m - 2)’; the extra common factors are (m - 1) in the first expression and (m + 3) in the 2nd expression.

Therefore, the required L.C.M. = m × (m - 2) × (m - 1) × (m + 3)

= m(m - 1) (m - 2) (m + 3)

Find the L.C.M. of x^{2} + xy, xz + yz and x^{2} + 2xy + y^{2}.

Solution:

Factorizing x^{2} + xy by taking the common factor 'x', we get

x(x + y)

Factorizing xz + yz by taking the common factor 'z', we get

z(x + y)

Factorizing x^{2} + 2xy + y^{2} by using the identity (a + b)^{2}, we get

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

= (x + y)^{2}

= (x + y) (x + y)

Therefore, L.C.M. of x^{2} + xy, xz + yz and x^{2} + 2xy + y^{2} is xz(x + y) (x + y).

Find the H.C.F. and the L.C.M. of 1.20 and 22.5

Solution:

Given, 1.20 and 22.5

Converting each of the following decimals into like decimals we get;

1.20 and 22.50

Now, expressing each of the numbers without the decimals as the product of primes we get

120 = 2 × 2 × 2 × 3 × 5 = 2^{3} × 3 × 5

2250 = 2 × 3 × 3 × 5 × 5 × 5 = 2 × 3^{2} × 5^{3}

Now, H.C.F. of 120 and 2250 = 2 × 3 × 5 = 30

Therefore, the H.C.F. of 1.20 and 22.5 = 0.30 (taking 2 decimal places)

L.C.M. of 120 and 2250 = 2^{3} × 3^{2} × 5^{3} = 9000

Therefore, L.C.M. of 1.20 and 22.5 = 90.00 (taking 2 decimal places)

Find the H.C.F. and the L.C.M. of 0.48, 0.72 and 0.108

Solution:

Given, 0.48, 0.72 and 0.108

Converting each of the following decimals into like decimals we get;

0.480, 0.720 and 0.108

Now, expressing each of the numbers without the decimals as the product of primes we get

480 = 2 × 2 × 2 × 2 × 2 × 3 × 5 = 2^{5} × 3 × 5

720 = 2 × 2 × 2 × 2 × 3 × 3 × 5 = 2^{4} × 3^{2} × 5

108 = 2 × 2 × 3 × 3 × 3 = 2^{2} × 3^{3}

Now, H.C.F. of 480, 720 and 108 = 2^{2} × 3 = 12

Therefore, the H.C.F. of 0.48, 0.72 and 0.108 = 0.012 (taking 3 decimal places)

L.C.M. of 480, 720 and 108 = 2^{5} × 3^{3} × 5 = 4320

Therefore, L.C.M. of 0.48, 0.72, 0.108 = 4.32 (taking 3 decimal places)

What is the Lowest common multiple (L.C.M) of 18 and 24 by using division method?

Solution:

2 | 18,24 |

2 | 9,12 |

3 | 3,6 |

2 | 1,2 |

1,1 |

Lowest common multiple (L.C.M) of 18 and 24 = 2 × 2 × 3 × 2 = 24.

What is the lowest common multiple (L.C.M) of 21 and 49 by using prime factorization method?

Solution:

3 | 21 |

7 | 7 |

1 |

7 | 49 |

7 | 7 |

1 |

To find the LCM, multiply all prime factors. But the common factors are included only once.

21 = 3 × 7

49 = 7 × 7 × 1

The required lowest common multiple (L.C.M) of 21 and 49 = 98 = 3 × 7 ×7× 1 = 147

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