If a number x divides another number y exactly (without leaving any remainder), then x is a factor of y and y is a multiple of x.

or

Factors Set of numbers which exactly divides the given number

Mulitples Set of numbers which are exactly divisible by the given number

For example, If the number is 8, then {8, 4, 2, 1} is the set of factors, while {8, 16, 24, 32, …..} is the set of multiples of 8.

Common Multiple:

A common multiple of two or more numbers is a number which is completely divisible (without leaving remainder) by each of them.

For example: We can obtain common multiples of 3, 5 and 10 as follows:

à Multiples of 3 = {3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33 ….}

à Multiples of 5 = {5, 10, 15, 20, 25, 30, 35, 40…}

à Multiples of 10 = {10, 20, 30, 40……}

Therefore, common multiples of 3, 5 and 10 = {30, 60, 90, 120 ….}

Least Common Multiple (LCM)

The LCM of two or more given numbers is the least number to be exactly divisible by each of them.

For example, We can obtain LCM of 4 and 12 as follows

Multiples of 4 = {4,8,12,16,20,24,32,36,….}

Multiples of 12 = {12,24,36,48,60,72,……}

Common multiples of 4 and 12 = 12, 24, 36, ……

LCM of 4 and 12 = 12

Methods to calculate LCM:

There are two methods to find the LCM of two or more numbers which are explained below.

1. Prime Factorisation Method

Following are the steps to obtain LCM through prime factorisation method.

Step 1: Resolve the given numbers into their prime factors.

Step 2: Find the product of all the prime factors (with highest powers) that occur in the given numbers.

Step 3: This product of all the prime factors (with highest powers) is the required LCM.

Example 1: Find the LCM of 8, 12 and 15.

Soln.

Factors of 8 = 2*2*2 = 2^{3}

Factors of 12 = 2*2*3 = 2^{2}*3^{1}

Factors of 15 = 3*5 = 3^{1}*5^{1}

Here, the prime factors that occur in the given numbers are 2, 3 and 5 and their highest powers are 2^{3}, 3^{1} and 5^{1}.

Required LCM = 2^{3}*3^{1}*5^{1} = 8*3*5 = 120

2. Division Method:

Following are the steps to obtain LCM through division method.

Step 1: Write down the given numbers in a line, separating them by commas.

Step 2: Divide them by a prime numbers which exactly divides atleast any two of the given numbers.

Step 3: Write down the quotients and the undivided numbers in a line below the 1^{st}.

Step 4: Repeat the process until you get a line of numbers which are prime to one another.

Step 5: The product of all divisors and the numbers in the last line will be the required LCM.

Example 2: What will be the LCM of 15, 24, 32, 45?

Soln. LCM of 15, 24, 32 and 45 is calculated as

Required LCM = 2*2*2*3*5*4*3 = 1440

Note: Start division with the least prime number.

Common Factor:

A common factor of two or more numbers is that particular number which divides each of them exactlyh.

For example: We can obtain common factors of 12, 48, 54 and 63 as follows:

Factors of 12 = 12, 6, 4, 3, 2, 1

Factors of 48 = 48, 24, 16, 12, 8, 6, 4, 3, 2, 1

Factors of 54 = 54, 27, 18, 9, 6, 3, 2, 1

Factors of 63 = 63, 21, 9, 7, 3, 1

Common factors of 12, 48, 54 and 63 = 3

Highest Common Factor:

HCF of two or more numbers is the greatest number which divides each of them exactly. For example 6 is the HCF of 12 and 18 as there is no number greater than 6 that divides both 12 and 18. Similarly, 3 is the highest common factor of 6 and 9.

HCF is also known as ‘Highest common Divisor’ (HCD) and ‘Greatest Common Measure’ (GCM).

Methods to Calculate HCF:

There are two methods to calculate the HCF of two or more numbers which are explained below.

1. Prime Factorisation Method:

Following are the steps for calculating HCF through prime factorisation method.

Step 1: Resolve the given numbers into their prime factors.

Step 2: Find the product of all the prime factors (with least power) common to all the numbers.

Step 3: the product of common prime factors (with the least powers) gives HCF.

Example 3: Find the HCF of 24, 30 and 42.

Soln. Resolving 24, 30 and 42 into their prime factors,

Factors of 24 = 2*2*2*3* = 2^{3}*3^{1}

Factors of 30 = 2*3*5 = 2^{1}*3^{1}*5^{1}

Factors of 42 = 2*3*7 = 2^{1}*3^{1}*7^{1}

The product of common prime factors with the least power = 2^{1}*3^{1} = 6

So, HCF of 24, 30 and 42 = 6.

2. Division Method:

Following are the seeps to obtain HCF through division method.

Step 1: Divide the larger number by the division method.

Step 2: Divide the divisor by the remainder

Step 3: Repeat the step 2 till the remainder becomes zero. The last divisor is the required HCF.

Note: To calculate the HCF of more than two numbers, calculate the HCF of first two numbers then take the third number and HCF of first two numbers and calculate their HCF and so on. The resulting HCF will be the required HCF of numbers.

Example 4: Find the HCF of 26 and 455.

Soln.

26)455(17

26

——-

195

182

————-

13)26(2

26

——-

X

Required HCF = 13

Remember:

To find the HCF of given numbers you can divide the numbers by their lowest possible difference. If these numbers are divisible by this difference, then this difference itself is the HCF of the given numbers any other factor of this difference will be its HCF.

Example 5: Find the HCF of 30, 42 and 135

Soln. We can notice that the difference between 30 and 42 is less than difference between 135 and 42 or 30.

Difference between 30 and 42 is 12, but 12 does don’t divide 30, 42 and 135 completely.

Factors of 12 = 12, 6, 4, 3, 2, 1

Clearly, 3 is the highest factor, which divides all the three numbers completely.

Therefore, 3 is the HCF of 30, 42 and 135.

Method to calculate LCM and HCF fractions:

The LCM and HCF can be obtained by the following formulae.

=> LCM of fractions = LCM of numerators / HCF of denominators

=> HCF of fractions = HCF of numerators / LCM of denominators

Note:

=> All the fractions must be in their lowest terms. If they are not in their lowest terms, then conversion in the lowest form is required before finding the HCF or LCM.

=> The required HCF of two or more fractions is the highest fraction which exactly divides each of the fractions.

=> The required LCM of two or more fractions is the least fraction / integer which is exactly divisible by each of them.

=> The HCF of numbers of fractions is always a fraction but this is not true in case of LCM.

Example 6: Calculate the LCM of 72/250, 126/75 and 162/165.

## TNPSC – HCF & LCM

HCF & LCM:Factors and Multiples:If a number x divides another number y exactly (without leaving any remainder), then x is a factor of y and y is a multiple of x.

or

FactorsSet of numbers which exactly divides the given numberMulitplesSet of numbers which are exactly divisible by the given numberFor example, If the number is 8, then {8, 4, 2, 1} is the set of factors, while {8, 16, 24, 32, …..} is the set of multiples of 8.

Common Multiple:A common multiple of two or more numbers is a number which is completely divisible (without leaving remainder) by each of them.

We can obtain common multiples of 3, 5 and 10 as follows:For example:à Multiples of 3 = {3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33 ….}

à Multiples of 5 = {5, 10, 15, 20, 25, 30, 35, 40…}

à Multiples of 10 = {10, 20, 30, 40……}

Therefore, common multiples of 3, 5 and 10 = {30, 60, 90, 120 ….}

Least Common Multiple (LCM)The LCM of two or more given numbers is the least number to be exactly divisible by each of them.

For example, We can obtain LCM of 4 and 12 as follows

Multiples of 4 = {4,8,12,16,20,24,32,36,….}

Multiples of 12 = {12,24,36,48,60,72,……}

Common multiples of 4 and 12 = 12, 24, 36, ……

LCM of 4 and 12 = 12

Methods to calculate LCM:There are two methods to find the LCM of two or more numbers which are explained below.

1. Prime Factorisation MethodFollowing are the steps to obtain LCM through prime factorisation method.

Resolve the given numbers into their prime factors.Step 1:Find the product of all the prime factors (with highest powers) that occur in the given numbers.Step 2:This product of all the prime factors (with highest powers) is the required LCM.Step 3:Find the LCM of 8, 12 and 15.Example 1:Soln.

Factors of 8 = 2*2*2 = 2

^{3}Factors of 12 = 2*2*3 = 2

^{2}*3^{1}Factors of 15 = 3*5 = 3

^{1}*5^{1}Here, the prime factors that occur in the given numbers are 2, 3 and 5 and their highest powers are 2

^{3}, 3^{1}and 5^{1}.Required LCM = 2

^{3}*3^{1}*5^{1}= 8*3*5 = 1202. Division Method:Following are the steps to obtain LCM through division method.

Write down the given numbers in a line, separating them by commas.Step 1:: Divide them by a prime numbers which exactly divides atleast any two of the given numbers.Step 2Write down the quotients and the undivided numbers in a line below the 1Step 3:^{st}.Repeat the process until you get a line of numbers which are prime to one another.Step 4:The product of all divisors and the numbers in the last line will be the required LCM.Step 5:What will be the LCM of 15, 24, 32, 45?Example 2:Soln. LCM of 15, 24, 32 and 45 is calculated as

Required LCM = 2*2*2*3*5*4*3 = 1440

Note: Start division with the least prime number.

Common Factor:A common factor of two or more numbers is that particular number which divides each of them exactlyh.

We can obtain common factors of 12, 48, 54 and 63 as follows:For example:Factors of 12 = 12, 6, 4, 3, 2, 1

Factors of 48 = 48, 24, 16, 12, 8, 6, 4, 3, 2, 1

Factors of 54 = 54, 27, 18, 9, 6, 3, 2, 1

Factors of 63 = 63, 21, 9, 7, 3, 1

Common factors of 12, 48, 54 and 63 = 3

Highest Common Factor:HCF of two or more numbers is the greatest number which divides each of them exactly. For example 6 is the HCF of 12 and 18 as there is no number greater than 6 that divides both 12 and 18. Similarly, 3 is the highest common factor of 6 and 9.

HCF is also known as ‘Highest common Divisor’ (HCD) and ‘Greatest Common Measure’ (GCM).

Methods to Calculate HCF:There are two methods to calculate the HCF of two or more numbers which are explained below.

1. Prime Factorisation Method:Following are the steps for calculating HCF through prime factorisation method.

Resolve the given numbers into their prime factors.Step 1:Find the product of all the prime factors (with least power) common to all the numbers.Step 2:the product of common prime factors (with the least powers) gives HCF.Step 3:Example 3: Find the HCF of 24, 30 and 42.

Soln. Resolving 24, 30 and 42 into their prime factors,

Factors of 24 = 2*2*2*3* = 2

^{3}*3^{1}Factors of 30 = 2*3*5 = 2

^{1}*3^{1}*5^{1}Factors of 42 = 2*3*7 = 2

^{1}*3^{1}*7^{1}The product of common prime factors with the least power = 2

^{1}*3^{1}= 6So, HCF of 24, 30 and 42 = 6.

2. Division Method:Following are the seeps to obtain HCF through division method.

Divide the larger number by the division method.Step 1:Divide the divisor by the remainderStep 2:Repeat the step 2 till the remainder becomes zero. The last divisor is the required HCF.Step 3:To calculate the HCF of more than two numbers, calculate the HCF of first two numbers then take the third number and HCF of first two numbers and calculate their HCF and so on. The resulting HCF will be the required HCF of numbers.Note:Find the HCF of 26 and 455.Example 4:Soln.

26)455(17

26

——-

195

182

————-

13)26(2

26

——-

X

Required HCF = 13

Remember:To find the HCF of given numbers you can divide the numbers by their lowest possible difference. If these numbers are divisible by this difference, then this difference itself is the HCF of the given numbers any other factor of this difference will be its HCF.

: Find the HCF of 30, 42 and 135Example 5Soln. We can notice that the difference between 30 and 42 is less than difference between 135 and 42 or 30.

Difference between 30 and 42 is 12, but 12 does don’t divide 30, 42 and 135 completely.

Factors of 12 = 12, 6, 4, 3, 2, 1

Clearly, 3 is the highest factor, which divides all the three numbers completely.

Therefore, 3 is the HCF of 30, 42 and 135.

Method to calculate LCM and HCF fractions:The LCM and HCF can be obtained by the following formulae.

=> LCM of fractions = LCM of numerators / HCF of denominators

=> HCF of fractions = HCF of numerators / LCM of denominators

Note:

=> All the fractions must be in their lowest terms. If they are not in their lowest terms, then conversion in the lowest form is required before finding the HCF or LCM.

=> The required HCF of two or more fractions is the highest fraction which exactly divides each of the fractions.

=> The required LCM of two or more fractions is the least fraction / integer which is exactly divisible by each of them.

=> The HCF of numbers of fractions is always a fraction but this is not true in case of LCM.

Calculate the LCM of 72/250, 126/75 and 162/165.Example 6:Here, 72/250 = 36/125, 126/75 = 42/25 and 162/165 = 54/55.Soln.According to the formula,

Required LCM = LCM of 36, 42 and 54 / HCF of 125, 25 and 55 = 756/5 = 151 1/5.

Find the HCF of 36/51 and 3 9/17.Example 7:Here, 36/51 = 12/17 and 3 9/17 = 60/17Soln.Now, we have to find the HCF of 12/17 and 60/17

According to the formula,

HCF of fractions = HCF of numerators / LCM of denominators

= HCF of 12 and 60 / LCM of 17 and 17 = 12/17

Note:The LCM and HCF of decimals can also be obtained from the above formulae (by converting the decimal into fraction).

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