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Lowest Common Multiple

The Lowest Common Multiple (LCM) of two or more positive integers is the smallest positive integer that is perfectly divisible by each of the given numbers. According to the Fundamental Theorem of Arithmetic, it is calculated as the product of the highest powers of all prime factors involved in the numbers.

Key Concepts

To find the LCM using the prime factorization method, first express each integer as a product of its prime factors.

Identify every unique prime factor present across all the given numbers.

Multiply the highest power of each of these prime factors together to obtain the final LCM.

Formula & Equation

LCM(a, b) * HCF(a, b) = a * b

a and b are two positive integers, LCM is their Lowest Common Multiple, and HCF is their Highest Common Factor.

Common Misconceptions

Myth: The LCM of two numbers is always calculated by simply multiplying the two numbers together.

Fact: The LCM is only equal to the product of the two numbers if they are co-prime, meaning they share no common factors other than 1. If they share factors, the LCM will be smaller than their product.

Real World Applications

Calculating the exact time when two or more traffic lights changing at different intervals will change simultaneously again.

Finding the minimum distance two people with different step lengths must walk so that both cover the exact same distance in complete steps.

Frequently Asked Questions

How do you find the LCM of two numbers using prime factorization?

Find the prime factors of both numbers, then multiply the highest power of each unique prime factor present in the expressions.

What is the relationship between LCM and HCF in Class 10 math?

For any two positive integers, the product of their LCM and HCF is exactly equal to the product of the two numbers themselves.

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