HCF AND LCM

In this section, we will learn HCF and LCM concepts and methods on how to find HCF and LCM.

Let us learn the terms that we are going to use.

  1. Factors: When one number, suppose “a” divides the other number “b” exactly, then a is called the factor of b.
    Also, “b” is called the multiple of “a”.
    For example, the number 42 is exactly divisible  by 6 and 7.
          Then, 6 and 7 are the factors of 42.
           and 42 is the multiple of 6 and 7.
  1. Common Factor: A common factor of two or more numbers, is the number that divides each of them exactly.
         For example,  5 is a common factor of 10, 25, and 125 .
  1. Common multiple: When two or more numbers have common numbers as their multiples, then those multiples are said to be common multiples.
      Example: 36, 12,24 are common multiples of 3 and 4.

LEAST OR LOWEST COMMON MULTIPLE( LCM )

“The LCM of two or more numbers is the least number which is exactly divisible by both of them.”
Example:
  15 is the least common multiple of 3 and 5.
Because, the multiple of 3 = 3,6,9,12,15,18,21,…
the multiple of 5 = 5,10,15,20,25,…

Methods of finding LCM of numbers

  1. METHOD OF PRIME FACTORISATION

In this method, we will express the given numbers into the product of the primes. Then the product of the highest powers of all the factors gives us the lcm.

Let us see that with an example,

Question: Find the Lcm of 12, 24, and 40.
Solution: Here,
12 = 2 × 2 × 3 = 22 × 3
24 = 2 × 2 × 2 × 3 = 23 × 3
40 = 2 × 2 × 2 × 5 = 23 × 5

Thus the Lcm of 12,24 and 40 = 22× 3 × 5 = 120.

Question: Find the Lcm of 12, 15, and 21.
Solution: Here,
12 = 2 × 2 × 3 = 22 × 3
15 = 3 × 5
21 = 3 × 7

Thus the Lcm of 12,15 and 21 = 22× 3 × 5× 7 = 420.

  1. BY COMMON DIVISION METHOD:
    ⇒Arrange the given numbers in a row, separated by commas.
    ⇒Now, divide by a number which divides at least two of the given numbers and carry forward the numbers which are not divisible.
    ⇒Repeat the process until no two numbers are divisible by the same number except
    1.
    ⇒The product of the divisors and the undivided numbers is the required Lcm.

 Let us take an example.

Question: Find the LCM of 12, 18, 36, and 48.
Solution: 


Thus LCM of 12, 18, 36 and 48 is = 2×2×2×2×3×3 = 144

Question: Find the lcm of 16, 24, 36, and 54.
Solution:


Thus the lcm of 16,24,36 and 54 is = 2×2×2×2×3×3×3 = 432

LCM OF FRACTIONS

To find the lcm of fractions – First, express the fraction in its lowest form.
Then

LCM OF FRACTIONS
LCM =\frac { LCM\quad of\quad numerator }{ HCF\quad of\quad denominator } 

Example:

Question:The LCM of \frac { 1 }{ 2 } and \frac { 5 }{ 8 } is?

Solution:
LCM = \frac { LCM\quad of\quad 1\quad and\quad 5 }{ HCF\quad of\quad 2\quad and\quad 8 }  = \frac { 5 }{ 2 } 

QUESTION : Find the LCM of  \frac { 2 }{ 3 } \frac { 4 }{ 9 } and \frac { 5 }{ 6 }.
Solution:
LCM =\frac { LCM\quad of\quad 2,4\quad and\quad 5 }{ HCF\quad of\quad 3 , 9\quad and\quad 6 }  =\frac { 20 }{ 3 }

LCM OF DECIMALS

⇒To find the lcm of decimals, make the same number of decimal places in each number by annexing zeroes in some numbers when necessary.
⇒Then find the Lcm of the numbers(as integers), considering them without decimals.
⇒Then mark the resultant LCM with as many decimals places as there were in each number.

Example:

Question: Find the Lcm of 0.6, 9.6, and 0.36.
Solution: First we will make there decimal places equal.
That is 0.60, 9.60, and 0.36.
Now find the lcm by removing the decimal.
We will find the lcm of 60,960 and 36.
The Lcm of 60, 960, and 36 is 2880.
Now, put the decimal in the resultant LCM with the same number of decimal places as there were in the numbers.

Thus, the LCM is= 28.80

HIGHEST COMMON FACTOR ( HCF)

The HCF of two or more numbers is the greatest number that divides each of them exactly.
The HCF is also called the highest common divisor or greatest common factor.

Example: HCF of 18,24 and 42 is 6. Because it is the largest number that divided all three of them exactly.

Methods of finding HCF of numbers

  1. METHOD OF PRIME FACTORISATION

In this method, we will express the given numbers into the product of the primes. Then find the product of all the prime factors common to all the numbers.

Let us see that with an example,

Question: Find the HCF of 24, 36, and 60.
Solution: Here,

24 = 2 × 2 × 2 × 3 = 23 × 3
36 = 2 × 2 × 3 × 3 = 22 × 32
60 = 2 × 2 × 3 × 5 = 22 × 3 × 5

Thus, the HCF of 12,24 and 60 = 22× 3 = 12.

  1. BY DIVISION METHOD:
    ⇒In this method, we will divide the greater number with a smaller number.
    ⇒If the remainder is not zero, we will divide the previous divisor with the remainder.
    ⇒Repeat this process of the division until the remainder is zero. The divisor for which the remainder becomes zero is the HCF of the numbers.

Let us take an example.

Question: Find the HCF of 63 and 105.

Solution:  105 > 63. so we divide 105 by 63.

Thus, the HCF of 105 and 63 is 21.

HCF OF MORE THAN TWO NUMBERS

To find the HCF of two or more numbers – First, we will find the HCF of any two numbers, then we will find the HCF of the other number and the HCF of previous numbers.

Example:
Question: Find the HCF of 513,1134 and 1215.
Solution: First we will find the HCF of 1134 and 1215.

Thus, the HCF of 1134 and 1215 is 81.

Now find the HCF of 81 and 513.

how to find hcf

Thus, HCF of 81 and 513 is 27.

Hence, HCF of 513, 1134, and 1215 is 27.

HCF OF FRACTIONS

To find the HCF of fractions – First, express the fraction in its lowest form.

Then,

HCF OF FRACTIONS
HCF = \frac { HCF\quad of\quad numerator }{ LCM\quad of\quad denominator }   

Example:

QUESTION : HCF of  \frac { 2 }{ 3 } \frac { 8 }{ 9 } \frac { 16 }{ 81 } and \frac { 10 }{ 27 } is?

Solution: HCF = \frac { HCF\quad of\quad 2,8,16\quad and\quad 10 }{ LCM\quad of\quad 3,9,81\quad and\quad 27 }  = \frac { 2 }{ 81 }

HCF OF DECIMALS

To find the HCF of decimals, make the same number of decimal places in each number by annexing zeroes in some numbers when necessary.
⇒Then find the HCF as integers of the numbers, considering them without decimals.
Then mark the resultant HCF with as many decimals places as there were in each number.

Example:

Question: Find the HCF of 0.63 ,1.05 and 2.1.
Solution: First we will make there decimal places equal.

That is 0.63, 1.05, and 2.10.

Now find the HCF by removing the decimal.

We will find the HCF of 63,105, and 210

The HC of 63, 105 and 210 is 21.

Now, put the decimal in the resultant HCF with the same number of decimal places as there were in the numbers.

Thus, the HCF is = 0.21

RELATIONSHIP BETWEEN HCF AND LCM

Relationship between HCF and LCM FOR TWO NUMBERS

Product of two numbers = Product of their HCF and LCM

Let the two positive integers be a and b.

Relationship between HCF and LCM FOR TWO NUMBERS
a × b = HCF( a, b) × LCM( a , b )

If the two numbers are given in ratios


If the two numbers are given in ratios, say x:y
If HCF is given.
Then,
First Number = HCF × x 
Second Number = HCF × y
Also,

LCM = HCF × x × y 
Relationship between HCF and LCM of THREE NUMBERS a, b and c

Relationship between HCF and LCM for three numbers a, b and c
HCF ( a , b , c ) = \frac { a\times b\times c\times LCM( a , b , c)  }{ LCM(a,b)\times LCM(b,c)\times LCM(a,c) }


Relationship between HCF and LCM for three numbers a, b and c
LCM (a,b,c)= \frac { a\times b\times c\times HCF(a,b,c)  }{ HCF(a,b)\times HCF(b,c)\times HCF(a,c) }
 

Co primes: Two or more numbers are said to be co-primes if their HCF is 1.

Example: (13,20) ,(7,17),(5,4)

To summarise,

how to find HCF and LCM of two numbers

Now that you have understood the concepts and formulas, let us look at the questions and answers, and understand how to find HCF and LCM. Click on the link below.

HCF and LCM questions and answers