Highest Common Factor (HCF) Calculator

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Highest Common Factor

A comprehensive guide about Highest Common Factor (GCD)

What is GCD?

The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), of two or more integers is the largest positive integer that divides each of the given numbers without leaving a remainder. GCD is a key concept in number theory and is widely used in simplifying fractions, solving Diophantine equations, and cryptography.

For example, the HCF or GCD of 12 and 18 is 6 because 6 is the largest number that divides both 12 and 18 evenly (12 ÷ 6 = 2, 18 ÷ 6 = 3).

Types of Ways to Find HCF or GCD

There are several methods to calculate the GCD of two or more numbers. Here, we’ll explore two common approaches: the Prime Factorization Method and the Division Method (also known as the ladder method).

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Prime Factorization Method

In this method, we break down each number into its prime factors, then take the lowest power of each prime that appears in all factorizations and multiply them to get the GCD.

Steps to Calculate HCF By Prime Factorization

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Example: Find the HCF of 25, 35 and 45.
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Prime factorization of 25: 25 = 5 x 5 = 52.
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Prime factorization of 35: 35 = 5 x 7 = 51 x 71.
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Prime factorization of 45: 45 = 5 x 9 = 51 x 91.
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Common primes with lowest powers: 5.
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GCD or HCF = 5.
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Division Method (Ladder Method)

In the division method, we divide the numbers by their common prime factors step-by-step until no further common division is possible, then multiply the divisors used to find the GCD.

Steps to Calculate LCM By Division Method

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Example: Find the HCF of 144 and 160.
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Start with 144 and 160.
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Divide by 2: 144 ÷ 2 = 72, 160 ÷ 2 = 80 → (72, 80).
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Divide by 2: 72 ÷ 2 = 36, 80 ÷ 2 = 40 → (36, 40).
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Divide by 2: 36 ÷ 2 = 18, 40 ÷ 2 = 20 → (18, 20).
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Divide by 2: 18 ÷ 2 = 9, 10 ÷ 2 = 10 → (9, 10).
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No further common divisors found. Stop dividing.
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Multiply all divisors: 2 × 2 × 2 × 2 = 16.
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HCF or GCD = 16.
Pro Tip

You can directrly divide 144, 160 to 16 because 16 is highest common divisor of 160 and 144.

Frequently Asked Questions (FAQ)

What is the HCF of two prime numbers?

The HCF of two prime numbers is always 1, as their only common factor is 1.

Can HCF be calculated for more than two numbers?

Yes, GCD can be calculated for any number of integers. Both methods (Prime Factorization and Division) can be extended to handle multiple numbers by applying the same logic iteratively.

What if one of the numbers is 0?

The LCM of 0 and any number is undefined in the context of positive integers because 0 does not have a meaningful set of multiples in this sense.

Which method is faster for large numbers?

The Division Method is often faster for larger numbers because it avoids the need to fully factorize each number into primes, especially if the numbers share many common factors.

How is HCF used in real life?

HCF can be used to split things into smaller sections, arrange items into groups, distribute items equally, and solve various problems related to division and sharing.

How does HCF relate to LCM?

HCF is the largest number that divides two or more numbers, while LCM is the smallest number that is a multiple of two or more numbers.