Finding the HCF of 162, 54, and 135

The Highest Common Factor (HCF) of a set of numbers is the largest positive integer that divides each of the numbers without leaving a remainder. We can find the HCF by using the prime factorization method.

Step-by-Step Calculation of HCF

Let's find the prime factorization for each number: 162, 54, and 135.

Prime Factorization of 162:

• Divide 162 by the smallest prime number, 2: \(162 \div 2 = 81\)
• 81 is not divisible by 2. Divide 81 by the next prime number, 3: \(81 \div 3 = 27\)
• Divide 27 by 3: \(27 \div 3 = 9\)
• Divide 9 by 3: \(9 \div 3 = 3\)
• Divide 3 by 3: \(3 \div 3 = 1\)

So, the prime factorization of 162 is \(2 \times 3 \times 3 \times 3 \times 3 = 2^1 \times 3^4\).

Prime Factorization of 54:

• Divide 54 by 2: \(54 \div 2 = 27\)
• Divide 27 by 3: \(27 \div 3 = 9\)
• Divide 9 by 3: \(9 \div 3 = 3\)
• Divide 3 by 3: \(3 \div 3 = 1\)

So, the prime factorization of 54 is \(2 \times 3 \times 3 \times 3 = 2^1 \times 3^3\).

Prime Factorization of 135:

• 135 is not divisible by 2. The sum of digits \(1+3+5=9\) is divisible by 3, so 135 is divisible by 3: \(135 \div 3 = 45\)
• Divide 45 by 3: \(45 \div 3 = 15\)
• Divide 15 by 3: \(15 \div 3 = 5\)
• Divide 5 by 5: \(5 \div 5 = 1\)

So, the prime factorization of 135 is \(3 \times 3 \times 3 \times 5 = 3^3 \times 5^1\).

Identifying Common Prime Factors and Calculating HCF

Now we list the prime factorizations of 162, 54, and 135:

• \(162 = 2^1 \times 3^4\)
• \(54 = 2^1 \times 3^3\)
• \(135 = 3^3 \times 5^1\)

To find the HCF, we look for the common prime factors present in all three factorizations and take the lowest power of each common factor.

• The prime factor 2 is present in 162 and 54, but not in 135. So, 2 is not a common factor to all three numbers.
• The prime factor 3 is present in 162 (\(3^4\)), 54 (\(3^3\)), and 135 (\(3^3\)). The lowest power of 3 that appears in all three factorizations is \(3^3\).
• The prime factor 5 is present only in 135. So, 5 is not a common factor to all three numbers.

The only common prime factor is 3, and its lowest power is \(3^3\).

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

HCF\((162, 54, 135) = 3^3 = 3 \times 3 \times 3 = 27\).

The common prime factor is 3, and the minimum exponent is 3.

HCF = \(3^3 = 27\).

Comparing this result with the given options:

• Option 1: 3
• Option 2: 1
• Option 3: 9
• Option 4: 27

Our calculated HCF, 27, matches Option 4.

Revision Table: Key Concepts for HCF

Additional Information on Finding HCF

Another method to find the HCF is the long division method, also known as the Euclidean algorithm. This method is particularly useful for finding the HCF of two larger numbers. To find the HCF of three numbers using the division method, you first find the HCF of any two numbers, and then find the HCF of the result and the third number.

For example, to find HCF(162, 54, 135):

• Find HCF(162, 54). \(162 = 3 \times 54 + 0\). The remainder is 0, so HCF(162, 54) = 54.
• Now find HCF(54, 135).
  • \(135 = 2 \times 54 + 27\)
  • \(54 = 2 \times 27 + 0\)
• The remainder is 0, so HCF(54, 135) = 27.

Therefore, HCF(162, 54, 135) = 27.

Both prime factorization and the division method will give you the same HCF for a set of numbers. Understanding HCF is important in simplifying fractions and solving problems involving distribution into equal groups.