Highest Common Factor (HCF)

The Highest Common Factor (HCF) is a key concept in mathematics that helps identify the largest number that can exactly divide two or more numbers without leaving a remainder. It’s widely used in simplifying fractions, solving problems involving ratios, and understanding number relationships.

The Highest Common Factor (HCF), is the largest positive integer that divides two or more numbers without leaving a remainder.

Example:
The HCF of 12 and 15 is 3, as 3 is the largest positive integer divisible by both 12 and 15 without leaving a remainder.

The prime factorization method involves breaking down each number into its prime factors and identifying the common factors.

Steps:
1. Write the prime factorization of each number.
2. Identify the common prime factors.
3. Multiply the common prime factors to get the HCF.

Example:
Find the HCF of 24 and 36

Prime factorization of 24:


Prime factorization of 36:


Common prime factors:
(as both have at least two 2’s) and

Multiply the common factors:


The HCF of 24 and 36 is 12.

The Highest Common Factor (HCF) is always a positive number in practical mathematics, as it represents the largest positive integer that divides two or more numbers without leaving a remainder. Negative numbers can also have common factors, but the HCF is conventionally taken as positive.

The HCF of monomials is the greatest monomial that divides all the given monomials exactly. It is found by determining the largest common factor of their coefficients and the smallest powers of all the variables common to the monomials.

Steps to Find the HCF of Monomials:

1. Factorize the coefficients: Find the highest common factor (HCF) of the numerical coefficients.

2. Compare variables: For each variable that appears in all monomials, take the smallest exponent.

3. Combine: Multiply the HCF of the coefficients with the variables and their smallest exponents.


Example:
Find the HCF of and

1. Factorize the coefficients:



HCF =

2. Compare the variables:

For : Take (smallest of and )
For : Take (smallest of and )

3. Combine the results:



The HCF =

Finding the HCF of binomials involves identifying the largest algebraic expression that divides each binomial exactly. Here’s a step-by-step explanation, followed by examples:

Steps to Find the HCF of Binomials:

1. Factorize each binomial completely: Break each binomial into its irreducible factors (e.g., common numerical coefficients, common variables with powers, or binomial factors).

2. Identify common factors: Look for the factors common to all the binomials.

3. Select the HCF: The HCF is the product of all the factors common to the binomials.


Example:
Find the HCF of and

1. Factorize each binomial:





2. Compare the variables:

Common numerical factor: 3 (from 6 and 3)

Common variable factor:

3. HCF:

Multiply the common factors:

Thus, the HCF =

• Real Numbers
• Whole Numbers
• Integers
• Rational Numbers
• Irrational Numbers

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