HCF And LCM Methods: Formula, Practice Questions And Answers

Exploring the realm of mathematics, the concepts of Highest Common Factor (HCF) and Lowest Common Multiple (LCM) play crucial roles in problem-solving and arithmetic operations.

Understanding these fundamental principles is key to simplifying fractions, solving equations, and tackling complex mathematical challenges with precision.

Defining HCF And LCM

HCF stands for Highest Common Factor and refers to the largest number that divides two or more numbers without leaving a remainder. The HCF is used when simplifying fractions, finding equivalent fractions, or reducing ratios. 

LCM stands for Lowest Common Multiple and refers to the least common denominator that signifies the smallest multiple, which is a common multiple of two or more numbers. LCM comes into play when adding or subtracting fractions with different denominators or solving problems related to periodic events like time intervals or cycles.

Calculation Methods Of HCF And LCM

Let us understand how to find LCM and HCF, i.e the different calculation methods:

Division Method Of HCF

When determining the HCF using the division method, you repeatedly divide the larger number by the smaller number until you get a remainder of zero. The last divisor before reaching zero is the HCF.

For example, divide 36 by 24, which gives us a quotient of 1 and a remainder of 12. Next, divide 24 by 12. It gives us a quotient of 2 and a remainder of 0. Since we reached a remainder of zero, the last divisor used, which is 12, is the HCF of 24 and 36.

Calculation Method Of LCM through Division

To find LCM through division, multiply the two numbers and divide them by their HCF to get the LCM. This method is efficient and straightforward for both HCF and LCM calculations.

For example, if we want to find the LCM of 12 and 18 using the division method, first, we calculate the HCF of 12 and 18, which is 6. Then, we multiply the two numbers (12 * 18 = 216) and divide them by their HCF (216 / 6 = 36). Hence, 36 is the LCM of 12 and 18. This method simplifies the process of finding both the HCF and LCM efficiently.

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HCF And LCM By Prime Factorization Method 

HCF

To find the HCF using prime factorization, each number is expressed as a product of its prime factors. Then, identify the common prime factors among the numbers.

Next, multiply these common prime factors to obtain the HCF of the given numbers. This method simplifies complex calculations and ensures accuracy in determining the highest common factor.

For instance, consider finding the HCF of 24 and 36. By listing their prime factors (24 = 2 x 2 x 2 x 3) and (36 = 2 x 2 x 3 x 3), we identify that the common factors are 2 and 3. Multiplying these gives us an HCF of 12.

LCM 

To find the least common multiple (LCM) using prime factorization, you first need to list down the prime factors of each number. Then, identify the highest power of every prime factor present in any of the numbers. Finally, multiply these highest powers together to get the LCM.

For instance, consider finding the LCM of 12 and 18. Here, we have 2 and 3 as the prime factors of 12 and 18, respectively. By selecting the highest power of each prime factor (2^2 * 3), we get an LCM of 12.

Using the prime factorization method to calculate HCF and LCM is essential as it provides a systematic and efficient way to find the highest common factor and the least common multiple of numbers. This method breaks down numbers into their prime factors, making it easier to identify the common factors and multiples, ultimately simplifying the calculation process.

Difference Between HCF And LCM

Let us study some of the specific differences between HCF and LCM:

HCF & LCM Practice Questions With Answers

Practice is the key to excelling in HCF and LCM! Provided below are some important selected questions with detailed answers:

Question 1: What is the highest common factor of 24 and 36?

a) 4

b) 8

c) 12

d) 16

Solution: c) 12

Explanation: 24=2×2×2×3
36=2×2×3×3
HCF=2×2×3
HCF=12

Question 2: Calculate the least common multiple of 9 and 15.

a) 15

b) 27

c) 45

d) 135

Solution: c) 45

Explanation: 9=3×3
15=3×5
LCM=3×3×5
LCM=45

Question 3: Calculate the HCF of 28 and 42.

a) 2

b) 6

c) 14

d) 84

Solution: c) 14

Explanation: 28=2×2×7
42=2×3×7
HCF=2×7
HCF=14

Question 4: What is the smallest positive integer that is a multiple of 127, has a prime number of positive divisors, and has a sum of digits equal to 10?

a) 101

b) 127

c) 152

d) 169

Solution: b) 127

Explanation: (i) The number 127 is a multiple of 127 itself

(ii) The number 127 has two positive divisors = 1 and 127, and "2" is a prime number

(iii) The sum of integers of 127 is 1+2+7=10.

Question 5: If the product of two numbers is 4576 and their HCF is 16, what is their LCM?

a) 143

b) 286

c) 572

d) 1144

Solution: b) 286

Explanation: Product of numbers=HCF×LCM
4576=16×LCM
LCM=286

Question 6: The product of three distinct positive integers is 120. The sum of their reciprocals is 47/20. What is the sum of the three integers?

a) 141

b) 26

c) 27

d) 28

Solution: a) 141

Explanation: a×b×c=120, 1/a+1/b+1/c=47/20
a+b+c=141

Question 7: The HCF of two numbers is 42. If their LCM is 2520 and one of the numbers is 630, what is the other number?

a) 105

b) 126

c) 210

d) 168

Solution: d) 168

Explanation: Product of numbers=HCF×LCM
a×630=42×2520
a=105840/630
a=168

Question 8: Find the sum of all positive integers less than 1000 that have an odd number of factors.

a) 1,960

b) 3,936

c) 10,416

d) 15,708

Solution: c) 10,416

Explanation: A positive integer has an odd number of factors if and only if it is a perfect square.
The largest perfect square less than 1000 is 961. Therefore sum of all numbers having an odd number of factors is
=1^2+2^2+3^2 -------- 31^2
=n(n+1)(2n+1)/6 .......... where n=31
=31×32×63/6
=62,496/6
=10,416

Question 9: Find the number of positive divisors of the number 2⁵×3⁴×5³×7^2.

a) 360

b) 720

c) 800

d) 900

Solution: a) 360

Explanation: If the prime factorization of a number is p1^n1​​×p2^n2​​×p3^n3​​×…×pk^nk, where p1,p2,…, pk​ are distinct primes and n1,n2,…,nk​ are their respective powers, then the number of positive divisors of the number is (n1​+1)(n2​+1)(n3​+1)…(nk​+1).
Number of positive divisors = (5+1)×(4+1)×(3+1)×(2+1)
=6×5×4×3
=360

Question 10: Find the LCM of 8 and 12.

a) 16

b) 24

c) 32

d) 48

Solution: b) 24

Explanation: 8=2×2×2
12=2×2×3
LCM=2×2×2×3
LCM=24

Conclusion

The comprehensive exploration of HCF and LCM has shed light on their definitions, calculation methods, and practical applications. By delving into prime factorization and division methods for both concepts, a solid foundation has been laid for understanding their significance in mathematics.

The examples provided have illustrated how these principles are applied in real-world scenarios, emphasizing the relationship between HCF and LCM. Moving forward, readers are encouraged to tackle the practice problems to solidify their understanding and mastery of these concepts.

Frequently Asked Questions (FAQs)

1. Give the HCF and LCM full form and explain their significance in mathematics. 

Understanding HCF (Highest Common Factor) and LCM (Least Common Multiple) is crucial in simplifying fractions, solving word problems, and analyzing relationships between numbers. It forms the foundation for various mathematical operations and helps in problem-solving strategies.

2. How are HCF and LCM calculated using prime factorization?

To find HCF by prime factorization, list the prime factors common to all numbers with the lowest power. For LCM, multiply all unique prime factors with their highest powers present in any number. The prime factorization method simplifies calculations and ensures accuracy in determining HCF and LCM.

3. What is the division method for calculating HCF?

The division method involves dividing two numbers successively to find their remainder until the remainder becomes zero. The divisor at this point is the HCF of those two numbers. This method provides a systematic approach to determine the highest common factor efficiently.

4. Why are examples and applications important when learning about HCF and LCM?

Examples and applications help reinforce understanding by demonstrating real-world scenarios where knowledge of HCF and LCM is applied. By practicing with diverse examples, learners can grasp concepts better, improve problem-solving skills, and appreciate the practical relevance of these mathematical principles.

5. How does understanding the relationship between HCF and LCM benefit problem-solving skills?

Recognizing the connection between HCF and LCM enables efficient problem-solving strategies by leveraging their properties. Understanding that LCM is a multiple of any number's factors while HCF divides them allows for quicker identification of solutions in various mathematical contexts.

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