Understanding HCF (Highest Common Factor)

The Highest Common Factor (HCF) of a set of numbers is the largest positive integer that divides each number in the set without leaving a remainder. It is also known as the Greatest Common Divisor (GCD).

Finding the HCF helps in simplifying fractions and solving problems related to equal distribution.

Methods to Find HCF

There are several methods to find the HCF of numbers. Two common methods are:

• Prime Factorization Method: Find the prime factors of each number and identify the common factors. The HCF is the product of the lowest powers of these common factors.
• Division Method: Use the division algorithm repeatedly to find the HCF of two numbers, then find the HCF of the result and the next number, and so on.

Calculating the HCF of 60, 148, and 382 using Prime Factorization

Let's use the prime factorization method to find the HCF of 60, 148, and 382 step-by-step.

Step 1: Find the prime factorization of each number.

We break down each number into its prime factors:

• Prime factorization of 60: $\qquad 60 = 2 \times 30 = 2 \times 2 \times 15 = 2^2 \times 3^1 \times 5^1$
• Prime factorization of 148: $\qquad 148 = 2 \times 74 = 2 \times 2 \times 37 = 2^2 \times 37^1$
• Prime factorization of 382: $\qquad 382 = 2 \times 191 = 2^1 \times 191^1$ (Note: 191 is a prime number)

We can summarize the prime factorizations in a table:

Step 2: Identify common prime factors.

Look for the prime factors that appear in the factorization of all three numbers (60, 148, and 382).

• The prime factor 2 is present in the factorization of 60 ($2^2$), 148 ($2^2$), and 382 ($2^1$).
• The prime factor 3 is only present in 60.
• The prime factor 5 is only present in 60.
• The prime factor 37 is only present in 148.
• The prime factor 191 is only present in 382.

The only common prime factor among 60, 148, and 382 is 2.

Step 3: Find the lowest power of each common prime factor.

For the common prime factor 2, find the smallest exponent it has across all the numbers:

• In 60, the power of 2 is 2 ($2^2$).
• In 148, the power of 2 is 2 ($2^2$).
• In 382, the power of 2 is 1 ($2^1$).

The lowest power of the common prime factor 2 is 1 ($2^1$).

Step 4: Calculate the HCF.

The HCF is the product of the lowest powers of all the common prime factors.

In this case, the only common prime factor is 2, and its lowest power is $2^1$.

HCF $= 2^1 = 2$

Conclusion: HCF of 60, 148, and 382

The Highest Common Factor (HCF) of 60, 148, and 382 is 2.

Revision Table: Key Concepts for HCF Calculation

Additional Information: HCF Applications and Properties

The concept of HCF is a fundamental building block in number theory and has practical applications:

• Simplifying Fractions: To reduce a fraction to its simplest form, divide both the numerator and the denominator by their HCF.
• Problem Solving: HCF is useful in real-world scenarios involving dividing quantities into equal parts or arranging items in rows or groups of the largest possible size.
• Relationship with LCM: For any two positive integers 'a' and 'b', the product of their HCF and Least Common Multiple (LCM) is equal to the product of the numbers: $\qquad \text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b$

Understanding how to find the HCF strengthens skills in arithmetic and algebraic manipulations.