Understanding the Highest Common Factor (HCF)

The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), of two or more numbers is the largest positive integer that divides into each of the numbers without leaving a remainder. To find the HCF of 42, 63, and 105, we can use the prime factorization method.

Prime Factorization Method to Find HCF

This method involves finding the prime factors of each number and then identifying the common factors.

Step 1: Find the prime factors of each number.

• For 42:

  \(42 = 2 \times 21 = 2 \times 3 \times 7\)

• For 63:

  \(63 = 3 \times 21 = 3 \times 3 \times 7 = 3^2 \times 7\)

• For 105:

  \(105 = 5 \times 21 = 5 \times 3 \times 7 = 3 \times 5 \times 7\)

Step 2: Identify the common prime factors.

Looking at the prime factorizations:

• \(42 = 2 \times \mathbf{3} \times \mathbf{7}\)
• \(63 = \mathbf{3^2} \times \mathbf{7}\)
• \(105 = \mathbf{3} \times 5 \times \mathbf{7}\)

The prime factors common to all three numbers are 3 and 7.

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

• The prime factor 3 appears with powers \(3^1\) in 42, \(3^2\) in 63, and \(3^1\) in 105. The lowest power is \(3^1\).
• The prime factor 7 appears with powers \(7^1\) in 42, \(7^1\) in 63, and \(7^1\) in 105. The lowest power is \(7^1\).

Step 4: Multiply the common prime factors raised to their lowest powers.

HCF = \(3^1 \times 7^1 = 3 \times 7 = 21\)

Result

The HCF of 42, 63, and 105 is 21.

This means 21 is the largest number that can divide 42, 63, and 105 without leaving any remainder.

• \(42 \div 21 = 2\)
• \(63 \div 21 = 3\)
• \(105 \div 21 = 5\)

Revision Table: HCF Concepts

Additional Information: Understanding HCF and GCD

HCF is often referred to as the Greatest Common Divisor (GCD). These terms mean the same thing. Finding the HCF is a fundamental concept in number theory and is useful in simplifying fractions and solving problems involving division.

Another method to find the HCF is the Euclidean Algorithm (Division Method), especially useful for larger numbers. For finding the HCF of 42, 63, and 105 using this method:

• First, find HCF(63, 105):

  \(105 = 1 \times 63 + 42\)

  \(63 = 1 \times 42 + 21\)

  \(42 = 2 \times 21 + 0\)

  HCF(63, 105) is 21.

• Then, find HCF(HCF(63, 105), 42), which is HCF(21, 42):

  \(42 = 2 \times 21 + 0\)

  HCF(21, 42) is 21.

Both methods confirm that the HCF of 42, 63, and 105 is 21.