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The HCF is the highest common factor of given two or more numbers and LCM is the Least Common Multiple of given two or more numbers. HCF is used to find less possible numbers and LCM is used to find a large number of given numbers. Both have numerous applications in maths simplification kinds of problems. There is also some relation between Highest Common Factor(HCF) and Least Common Multiple(LCM).
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Read Also: Class 12 Euclid Division Lemma
Highest Common Factor:
HCF, also known as GCD, is the Highest Common Factor of two or more numbers. It is the largest positive integer that can divide the given numbers without remainders. Let’s look at an example,
Example: Find the Highest Common Factor(HCF) of 20 and 36
Solution:
The factors, f of 20 are 1, 2, 4, 5, 10, and 20
The Factors, f of 36 are 1, 2, 3, 4, 6, 12, 18, and 36
The Common Factors, f of 20 and 36 are 1, 2, and 4.
The Largest among the common factor is 4
Therefore the Highest Common Factor(HCF) of 20 and 36 is 4,
Methods to find HCF:
There are two methods to find the Highest Common Factor of given numbers, they are
- Division Method
- Prime Factorization Method
Division Method:
To find the HCF of two or more given numbers, follow the below-mentioned tips
Step 1: Draw a long L shaped line and write the given numbers separated by a comma
Step 2: On the left side of the L shaped line, write the smallest prime factor which divides the given numbers without any remainders.
Step 3: Then divide the given numbers with the prime number and write down the quotient under the lines.
Step 4: Repeat the step 1 to 3 till there is no more common prime factor for given numbers.
Step 5: The prime factor on the left side is the common prime factor for given numbers, so write it down separately.
Step 6: To find HCF using the obtained common prime factors, f, simply multiply the common prime factors, f.
Read More: Class 10 Real Numbers
Example for Division Method:
- Find the HCF of 16 and 36 using the long division method?
Solution: The given two numbers are 16 and 36, we have to find HCF of these two numbers by long division method, so
The common prime factors, f are 2, 2
To find the Highest Common Factor multiply these common prime factors, f
ie 2 x 2 = 4
Therefore the Highest Common Factor(HCM) of 16 and 36 is 4
Prime Factorization Method:
This method is the easy method to find the HCF of two or more numbers, follow the steps to find HCF in the prime factorization method
Step1: Write down the given numbers separately, then write the factors, f of the numbers next to it
Step 2: Write down the common factors, f of given numbers(write only lowest degree factors, f)
Step 3: Find out the Highest Common Factors, f from the list of common factors, f.
The highest common factor is the HCF of the given numbers, look at the example below
Example of Prime Factorization Method:
- Find the Highest Common Factor of 21 and 28 by prime factorization method?
Solution:
The factors, f of 21 are, 1, 3, 7 and 21
The factors, f of 28 are 1, 2, 4, 7, 14 and 28
The Common factors, f of 21 & 28 are 1, 7
The highest common factor is 7
Therefore the Highest Common Factor(HCF) of 21 and 28 is 7
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Least Common Multiples
It is the smallest possible integer which is divided by all the given numbers with any remainders. The Least Common Multiple of given numbers is always greater than the given numbers, Let’s look at an example,
Example: Find out the Least Common Multiple of 16 and 12
The multiples of 16 are 16, 32, 48, 64, 80….. ∞ and the multiples of 12 are 12, 24, 36, 48, 60….. ∞
On comparing, the least common multiple of 16 and 12 is 48
Methods to Find LCM:
There are two methods to find the LCM of given numbers, They are
- Long Division method
- Prime Factorization method
Long Division Method:
This method is the same as the division method of HCF the only difference is we have to multiply the final quotient with the left side numbers. Let’s take a look at the example to understand the division method clearly.
Example of Long Division Method:
- Find the Least Common Multiple of 12 and 16 using the Long Division method.
Solution: To find the Least Common Multiple, multiply left side numbers 2, 2 with quotients 3, 4
This gives 2 x 2 x 3 x 4=48
Therefore by the division method Least Common Multiple(LCM) of 12 & 16 is 48
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Prime Factorization Method:
To find LCM using the Prime factorization method, follow the below-mentioned steps
Step 1: Write the given numbers separately, and write down the possible prime factors, of which multiplication gives the number.
Step 2: Take the highest occurrence prime factors, f from all the given numbers and single occurrence number.
Step 3: Multiply the highest occurrence prime factors, f and single occurrence prime factors, f of given numbers to find the LCM
Example of Prime Factorization Method
- Find the Least Common Multiple of 16 and 14 using the prime factorization method.
Solution:
The prime factors, f of 16 are 2 x 2 x 2 x 2
The prime factors, f of 14 are 2 x 7
In this example 2 is the common prime factor of 16 and 14 and it is the highest occurrence number(4 times in 16)
So To find LCM, multiply 2 four times and 7
It gives, 2 x 2 x 2 x 2 x 7 = 112
Therefore by the prime factorization method, the Least Common Multiple(LCM) of 14 and 16 is 112
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Relation Between HCF and LCM:
- The product of HCF and LCM of given numbers is always equal to the product of the two numbers.
If a and b are two given positive numbers, then a x b = HCF(a, b) x LCM(a, b)
Proof: If 8 and 10 are two numbers substitute a and b with 8 and 10 in an above-mentioned equation, we get
8 x 10 = HCF(8, 10) x LCM(8, 10)
RHS:
Calculate Highest Common Factor(HCF) of 8, 10 by prime factorization method
- The factors, f of 8 are 1, 2, 4 and the factors, f of 10 are 1, 2, 5
- The Highest Common Factor of 8 and 10 is 2
Calculate Least Common Multiple(LCM) of 8 and 10 by prime factorization method
- The prime factors, f of 8 are 2 x 2 x 2 and the prime factors, f of 10 are 2, 5
- The Least Common Multiple(LCM) of 8 and 10 is 40
Then HCF(8,10) x LCM(8, 10) = 2 x 40= 80
LHS:
8 x 10 = 80
LHS = RHS, Hence Proved
- The Least Common Multiple of the co-prime number is equal to the product of a given number because the HCF of the co-prime number is always 1
- The LCM and HCF are important to find the LCM and HCF of any fractions, The formulas of H.C.F. and L.C.M. of Fractions are
The formula for Least Common Multiple of fractions = LCM of Numerators / HCF of Denominators
The formula for Highest Common Factor of fractions = HCF of Numerators / LCM of Denominators
Points to Remember:
- The HCF is also known as Great Common Divisor (GCD) is the highest common factor of given numbers whereas the LCM is the Least Common Multiple of given numbers. Both the Division method and prime factorization method are used to find the HCF and LCM of given numbers
- The product of the Highest Common Factor(HCF) and Least Common Multiple(LCM) of two numbers is always equal to the product of two numbers.
- For coprime numbers, HCF is always one, so the LCF of a given coprime number is always equal to the product of the given numbers
- The HCF of the given numbers is always smaller than the given numbers and the LCM of given numbers is always larger than the given numbers
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Sample Questions
Ques. Find the Highest Common Factor(HCF) and Least Common Multiple(LCM) of 64, 72, 84 using the prime factorization method. (4 marks)
Solution: To find Highest Common Factor(HCF)
The factors, f of 64 are 1, 2, 4, 8, 16, 32, and 64
The factors, f of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72
The factors, f of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42 and 84 and the common factors, f are 1, 2, 4
Therefore the Highest Common Factor(HCF) of 64, 72 and 84 are 4
To Find Least Common Multiple (LCM),
The prime factors, pf of 64 are 2 x 2 x 2 x 2 x2 x 2
The prime factors, pf of 72 are 2 x 2 x 2 x 3 x 3
And The prime factors, pf of 84 are 2 x 2 x 3 x 7
LCM = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 7
The Least Common Multiple(LCM) of 64, 72 and 84 is 4032
Ques. Find the LCM and HCF of fractions 11/13, 21/28, 8/15 (5 marks)
Solution: To calculate the LCM and HCF of fractions, we have to use the formulas,
Least Common Multiple(LCM) of fractions = Least Common Multiple(LCM) of Numerators / Highest Common Factor(HCF) of Denominators
Highest Common Factor(HCF) of fractions = Highest Common Factor(HCF) of Numerators / Least Common Multiple(LCM) of Denominators
LCM of Fractions:
Least Common Multiple(LCM) of fractions = Least Common Multiple(LCM) of Numerators / Highest Common Factor(HCF) of Denominators
We have to calculate LCM of numerators and HCF of denominators, using the prime factorization method,
LCM of 11, 21, 8
Prime factors, pf of 11 are 1, 11
Prime Factors, pf of 21 are 1, 3, 7
Prime Factors, pf of 8 are 2, 2, 2
The Least Common Multiple(LCM) of 11, 21, 8 is 2 x 2 x 2 x 3 x 7 x 11 = 1848
HCF of 13, 28, 15
Factors, f of 13 are 1, 13
Factors, f of 28 are 1, 2, 4, 7, 14, 28
Factors, f of 15 are 1, 3, 5, 15
The HCF of 13, 28, 15 is 1
So the LCM of fractions = 1848/1 = 1848
Read More: Minors and Cofactors
HCF of Fractions:
Highest Common Factor of fractions = Highest Common Factor(HCF) of Numerators / Least Common Multiple(LCM) of Denominators
HCF of 11, 21, 8 is 1
LCM of 13, 28, 15
Prime Factors, pf of 13 are 1, 13
Prime Factors, pf of 28 are 1, 2, 2, 7
Prime Factors, pf of 15 are 1, 3, 5
The Least Common Multiple(LCM) is 2 x 2 x 3 x 5 x 7 x 13 = 5460
The Highest Common Factor of fractions is 1/5460
Therefore LCM of 11/13, 21/28, 8/15 is 1848 and HCF of 11/13, 21/28, 8/15 is 1/5460
Ques. The LCM and HCF of two natural numbers is 168 and 2 respectively. If one of the natural numbers is 24 what is another number? (2 marks)
Solution: We can use the following expression to find the another number
a x b = HCF(a, b) x LCM(a, b), where a and b are natural numbers
By substituting, a = 24, HCF = 2 and LCM = 168 we get
2 x 168 = 24 x b
b = (2 x 168)/24
b = 336/24
b = 14
Therefore the another natural number is 14
Ques. Find the HCF and LCM of 124 and 168, then prove that HCF x LCM = Product of Numbers. (3 marks)
Solution:
LHS:
To find HCF:
By Prime Factorization method,
The prime factors, pf of 124 are 2 x 2 x 31
The prime factors, pf of 168 are 2 x 2 x 2 x 3 x 7
So the Highest Common Factor(HCF) of 124 and 168 is 4
To find LCM,
By Prime Factorization Method,
The prime factors, pf of 124 are 2 x 2 x 31
The prime factors, pf of 168 are 2 x 2 x 2 x 3 x 7
So the Least Common Multiple(LCM) of 124 and 168 is 5208
So, HCF x LCM = 4 x 5208
Therefore the product of HCF and LCM is 20832
RHS:
Product of given numbers, 124 x 168 = 20832
LHS = RHS, Hence Proved
Read More: Class 12 Determinants
Ques. Find HCF and LCM of 112, 134, 204 by prime factorization method(pf method). (2 marks)
Solution: By prime factorization method,
The prime factors, pf of 112 are 2 x 2 x 2 x 2 x 7
The prime factors, pf of 134 are 2 x 67
The prime factors, pf of 204 are 2 x 3 x 17
So, the Highest common factor(HCF) of 112, 134, 204 is 2
And the Least Common Multiple(LCM) of 112, 134, 204 is 382704
Ques. Find the LCM of 404 and 702 whose HCF is 2. (1 mark)
Solution: We know that,
HCF x LCM = 404 x 702
So, LCM = (404 x 702)/2
Therefore LCM = 141804
Ques. A vegetable vendor has 48 cabbages, 72 carrots and 108 brinjals. He wants to arrange this in a row like he has to arrange only one type of vegetable in each row. What is the minimum number of rows he required to arrange all the vegetables? (3 marks)
Solution: Given, Cabbages = 48, Carrots = 72, Brinjals = 108
The prime factors, pf of 48 are 2 x 2 x 2 x 2 x 3
The prime factors, pf of 72 are 2 x 2 x 3 x 3
The prime factors, pf of 108 are 2 x 2 x 3 x 3 x 3
Minimum number of rows = Maximum number of vegetables in a row
The Highest common factor(HCF) of 48, 72, 108 is 12
Therefore Minimum rows = (48/12) + (72/12) + (108/12) = 4 + 6 + 9 = 19
The minimum total number of rows required to arrange the vegetables is 19
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Previous year Questions
Ques. Find the HCF and LCM of 404 & 96 and prove that HCF x LCM = 404 x 96 (2018)
Solution: Given numbers are 404 & 96
The prime factors, pf of 404 are 2 x 2 x 101
The prime factors, pf of 96 are 2 x 2 x 2 x 2 x 2 x 3
Therefore the Highest Common Factor(HCF) of 404 and 96 is 4
And Least Common Multiple(LCM) of 404 & 96 is 9696
LHS: HCF x LCM = 4 x 9696 = 38784
RHS: 404 x 96 = 38784
Hence Proved
Ques. The two positive integers p and q are written as p = a2b and q = ab2, where a and b are prime numbers. Prove that HCF(p,q) x LCM(p,q) = p x q. (2017)
Solution: Given p = a2b, q = ab2
Where a, b are prime numbers
LCM(p,q) = a x a x b x b
HCF(p,q) = a x b
So LHS = a3b3
RHS = a x a x b x a x b x b = a3b3
LHS = RHS, hence proved
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