How do you find the LCM using prime factorization?

To find the LCM, you first find the prime factorization of each number. Then, you take the highest power of every prime factor that appears in any of the factorizations and multiply them together.

How do you find the HCF using prime factorization?

To find the HCF, you first find the prime factorization of each number. Then, you take the lowest power of every prime factor that is common to all of the factorizations and multiply them together.

What is the relationship between LCM and HCF for two numbers?

For any two positive integers 'a' and 'b', the product of their LCM and HCF is equal to the product of the numbers themselves. The formula is: LCM(a, b) × HCF(a, b) = a × b. Our calculator demonstrates this relationship in the solution steps.

LCM and HCF Calculator

Find the Least Common Multiple (LCM) and Highest Common Factor (HCF/GCD) with visual, step-by-step solutions.

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Find the HCF (GCD)

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Understanding the Concepts

Prime Factorization

Think of prime numbers (2, 3, 5, 7...) as the fundamental 'building blocks' of all other numbers. Prime factorization is the process of breaking down a number into its unique set of prime building blocks.

Example: The prime factorization of 12 is 2 × 2 × 3. No other combination of prime numbers will ever multiply to 12. This unique 'recipe' is the key to finding both the LCM and HCF.

Least Common Multiple (LCM)

The LCM is the smallest positive number that is a multiple of all numbers in a set. Imagine two runners on circular tracks of different lengths; the LCM tells you the first time they will cross the starting line together again.

Use Case: When adding fractions like 1/4 and 1/6, you need a common denominator. The LCM of 4 and 6 (which is 12) provides the 'least common denominator', allowing you to solve the problem efficiently.

Highest Common Factor (HCF/GCD)

The HCF (also called GCD) is the largest positive number that divides into all numbers in a set without leaving a remainder. It's about finding the biggest common piece they share.

Use Case: To simplify a fraction like 48/60, you find the HCF of 48 and 60 (which is 12). Dividing both the top and bottom by 12 gives you 4/5, the fraction's simplest form.

Frequently Asked Questions

What is Prime Factorization?

Prime factorization is the process of breaking down a number into a product of its prime factors. For example, the prime factorization of 12 is 2 × 2 × 3. This is the fundamental method our calculator uses to find the LCM and HCF.

What's the difference between HCF and GCD?

There is no difference. "Highest Common Factor" (HCF) and "Greatest Common Divisor" (GCD) are two different names for the exact same concept. HCF is more common in the UK, while GCD is more common in the US.

How do you find the LCM of 12 and 18?

First, find the prime factors: 12 = 2² × 3 and 18 = 2 × 3². To find the LCM, take the highest power of each prime factor present in either number (2² and 3²) and multiply them: 2² × 3² = 4 × 9 = 36. So, the LCM is 36.

How do you find the HCF of 48 and 60?

First, find the prime factors: 48 = 2⁴ × 3 and 60 = 2² × 3 × 5. To find the HCF, take the lowest power of each prime factor that is common to both numbers (2² and 3) and multiply them: 2² × 3 = 4 × 3 = 12. So, the HCF is 12.