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, Show 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: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:
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:
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 Read More:
Least Common MultiplesIt 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: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:
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 Check Important Formulas for Logarithm 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
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 Read More:
Relation Between HCF and LCM:
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
Calculate Least Common Multiple(LCM) of 8 and 10 by prime factorization method
Then HCF(8,10) x LCM(8, 10) = 2 x 40= 80 LHS: 8 x 10 = 80 LHS = RHS, Hence Proved
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:
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 Check out Important Links For:
Previous year QuestionsQues. 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 Relatable Links: |
What is the relation between HCF and LCM of two numbers?
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