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Introduction
When multiple numbers have common multiples, the smallest common multiple among them is known as the least common multiple (LCM). The LCM formula helps in determining this smallest multiple for given numbers. In simpler terms, the LCM of two integers, denoted by a and b, is the smallest positive integer that is divisible by both a and b.
What is Least Common Multiple?
LCM, or the Least Common Multiple, is the smallest positive integer that is divisible by two or more numbers.
For Example, Consider two numbers, 4 and 6.
To find the LCM of 4 and 6, we list the multiples of each number: 4, 8, 12, 16, 20, 24… and 6, 12, 18, 24… The common multiples are 12 and 24, but the smallest common multiple is 12. Therefore, the LCM of 4 and 6 is 12.
In this example, 12 is the smallest positive integer that is divisible by both 4 and 6, making it their least common multiple.
LCM Formula
The LCM formula can be expressed as,
LCM Formula:
LCM = (a × b)/HCF(a,b)
Where LCM(a, b) represents the least common multiple of numbers a and b, and HCF(a, b) represents the highest common multiple of a and b.
By using the LCM formula, we can efficiently calculate the LCM of two numbers by first finding their HCF and then applying it in the formula.
Conclusion
The LCM formula is helpful in various scenarios, such as finding a common denominator for fractions, solving equations involving multiple variables, simplifying expressions, and working with fractions and ratios.
Understanding and applying the LCM formula allows us to find the smallest common multiple of numbers, which is essential in many mathematical and real-life situations where multiples or divisibility are involved.
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Solved Examples on LCM
Example 1: The product of two numbers is 240 and their highest common factor (HCF) is 12. Find their least common multiple (LCM).
Solution: Let’s assume the two numbers are a and b.
Given: a × b = 240
HCF(a, b) = 12
To find the LCM using the formula LCM(a, b) = (|a × b|) / HCF(a, b), we need to find the HCF (Highest Common Factor) first.
Since HCF(a, b) = 12, we know that 12 is a factor of both a and b.
Let’s write the prime factorization of 12: 12 = 2² × 3
To find the remaining prime factorization, we divide 240 by 12:
240 / 12 = 20
Now, let’s write the prime factorization of 20: 20 = 2² × 5¹
Combining the prime factorizations of 12 and 20, we get:
240 = 2² × 3¹ × 5¹
Therefore, the LCM of the two numbers is given by:
LCM(a, b) = (|a × b|) / HCF(a, b) = (|240|) / (2² × 3¹ × 5¹) = 240 / 12 = 20
Therefore, the LCM of the two numbers is 20.
Example 2: The product of two numbers is 72 and their least common multiple (LCM) is 24. Find their highest common factor (HCF).
Solution: Let’s assume the two numbers are a and b.
Given: a × b = 72
LCM(a, b) = 24
To find the HCF using the formula HCF(a, b) = (|a × b|) / LCM(a, b), we need to find the LCM (Least Common Multiple) first.
Since LCM(a, b) = 24, we know that 24 is a multiple of both a and b.
Let’s write the prime factorization of 24: 24 = 2³ × 3¹
To find the remaining prime factorization, we divide 72 by 24:
72 / 24 = 3
Now, let’s write the prime factorization of 3: 3 = 3¹
Combining the prime factorizations of 24 and 3, we get:
72 = 2³ × 3¹
Therefore, the HCF of the two numbers is given by:
HCF(a, b) = (|a × b|) / LCM(a, b) = (|72|) / (2³ × 3¹) = 72 / 24 = 3
Therefore, the HCF of the two numbers is 3.
Frequently Asked Questions on LCM
How to calculate the LCM?
To calculate the LCM (Least Common Multiple) of two or more numbers, you can use different methods. One common approach is to list the multiples of each number and find the smallest common multiple. Another method is prime factorization, where you express each number as a product of prime factors and then determine the LCM by taking the highest power of each prime factor. Additionally, you can also use the LCM formula, LCM(a, b) = |(a × b)| / HCF(a, b), where HCF represents the Highest Common Factor.
What is the formula of HCF and LCM?
The formula for the HCF (Highest Common Factor) of two numbers is obtained by dividing their product by their LCM (Least Common Multiple). Mathematically, it can be expressed as HCF(a, b) = (a × b) / LCM(a, b). This formula helps find the HCF when the LCM is known.
What is LCM formula?
The LCM formula is LCM(a, b) = |(a × b)| / HCF(a, b). It calculates the least common multiple of two numbers using their product and highest common factor.
What is LCM full form?
LCM stands for Least Common Multiple, which refers to the smallest positive integer that is divisible by both given numbers.
Is LCM a multiple of HCF?
No, LCM is not necessarily a multiple of HCF. The relationship between LCM and HCF depends on the numbers being considered. In some cases, the LCM and HCF can have a common factor, while in other cases they may be completely independent. It is important to calculate both the LCM and HCF separately to understand their specific properties and relationships in a given scenario.
What is the LCM of 24 and 36?
To find the LCM (Least Common Multiple) of 24 and 36, we can use the prime factorization method or the LCM formula. Prime factors of 24: 2³ × 3¹ Prime factors of 36: 2² × 3 To calculate the LCM, we take the highest powers of all the prime factors: LCM = 2³ × 3² = 8 × 9 = 72 Therefore, the LCM of 24 and 36 is 72.
What are the properties of LCM?
The LCM of co-prime numbers is always equal to the product of the numbers. Additionally, the LCM of any given numbers is always greater than or equal to any of the given numbers.
What is the full form of HCF?
The full form of HCF is Highest Common Factor, also known as Greatest Common Divisor (GCD).
