highest common factor

Highest Common Factor (HCF) in Math

Definition

The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), is the largest positive integer that divides two or more numbers without leaving a remainder. It represents the highest factor that is common to all the given numbers.

Knowledge Points

The concept of the Highest Common Factor involves the following knowledge points:

1. Factors: A factor of a number is an integer that divides the number without leaving a remainder.
2. Divisibility: A number is divisible by another number if it can be divided evenly without leaving a remainder.
3. Prime Numbers: Prime numbers are numbers greater than 1 that have no factors other than 1 and themselves.
4. Prime Factorization: Prime factorization is the process of expressing a number as a product of its prime factors.

Formula or Equation

The HCF of two or more numbers can be calculated using the prime factorization method. The formula for finding the HCF is:

HCF(a, b) = Product of common prime factors of a and b

Application of the HCF Formula

To find the HCF using the formula, follow these steps:

1. Find the prime factorization of each number.
2. Identify the common prime factors.
3. Multiply the common prime factors to obtain the HCF.

Symbol for Highest Common Factor

The symbol for the Highest Common Factor is "HCF" or "GCD" (Greatest Common Divisor).

Methods for Finding the HCF

There are several methods to find the HCF:

1. Prime Factorization Method: This method involves finding the prime factors of each number and identifying the common factors.
2. Division Method: This method involves dividing the larger number by the smaller number repeatedly until the remainder becomes zero. The last divisor used is the HCF.
3. Euclidean Algorithm: This algorithm involves repeatedly dividing the larger number by the smaller number and replacing the larger number with the remainder until the remainder becomes zero. The last non-zero remainder is the HCF.

Solved Examples

Example 1: Find the HCF of 24 and 36.

Solution: Step 1: Prime factorization of 24 = 2^3 * 3 Step 2: Prime factorization of 36 = 2^2 * 3^2 Step 3: Common prime factors = 2^2 * 3 = 12 HCF(24, 36) = 12

Example 2: Find the HCF of 45, 60, and 75.

Solution: Step 1: Prime factorization of 45 = 3^2 * 5 Step 2: Prime factorization of 60 = 2^2 * 3 * 5 Step 3: Prime factorization of 75 = 3 * 5^2 Step 4: Common prime factors = 3 * 5 = 15 HCF(45, 60, 75) = 15

Practice Problems

1. Find the HCF of 18 and 24.
2. Find the HCF of 72, 96, and 120.
3. Find the HCF of 63 and 81.

FAQ

Question: What is the highest common factor? Answer: The highest common factor, also known as the greatest common divisor, is the largest positive integer that divides two or more numbers without leaving a remainder. It represents the highest factor that is common to all the given numbers.