How to Find the HCF of 54, 126, and 144 Using Prime Factorization Method

In this article, we will explore how to find the Highest Common Factor (HCF) of 54, 126, and 144 using the prime factorization method. Prime factorization is a fundamental technique in number theory that can be used to find the HCF efficiently. This method is particularly useful for larger numbers and can provide a clear understanding of the factors involved.

Step 1: Prime Factorization of Each Number

To find the HCF using prime factorization, start by finding the prime factors of each number individually.

Prime Factorization of 54

54 is an even number, so we divide it by 2: 54 ÷ 2 27 27 is divisible by 3: 27 ÷ 3 9 9 is divisible by 3: 9 ÷ 3 3 3 is a prime number.

The prime factorization of 54 is: 54 21 × 33

Prime Factorization of 126

126 is an even number, so we divide it by 2: 126 ÷ 2 63 63 is divisible by 3: 63 ÷ 3 21 21 is divisible by 3: 21 ÷ 3 7 7 is a prime number.

The prime factorization of 126 is: 126 21 × 32 × 71

Prime Factorization of 144

144 is an even number, so we divide it by 2: 144 ÷ 2 72 72 is even, so we divide it by 2: 72 ÷ 2 36 36 is even, so we divide it by 2: 36 ÷ 2 18 18 is even, so we divide it by 2: 18 ÷ 2 9 9 is divisible by 3: 9 ÷ 3 3 3 is a prime number.

The prime factorization of 144 is: 144 24 × 32

Step 2: Identify Common Prime Factors

Now, we need to list the prime factors of each number:

54: 21 × 33 126: 21 × 32 × 71 144: 24 × 32

Step 3: Find the Lowest Power of Common Prime Factors

The common prime factors are 2 and 3. For the lowest power:

2: The lowest power is 21 3: The lowest power is 32

Step 4: Calculate the HCF

Multiplying the lowest powers of the common prime factors gives us the HCF:

HCF 21 × 32 2 × 9 18

Conclusion

The Highest Common Factor (HCF) of 54, 126, and 144 is 18.

Alternatively, we can also use a different method where we divide each number by the smallest prime factor (2) and then by the next smallest prime factor (3) until we can no longer do so. Here's a step-by-step breakdown:

54, 126, and 144 are all even numbers, so we divide each by 2: 54 ÷ 2 27, 126 ÷ 2 63, 144 ÷ 2 72 Next, we check if 27, 63, and 72 are multiples of 9 by adding their digits: 2 7 9, 6 3 9, 7 2 9, so they are all multiples of 9. We divide each by 9: 27 ÷ 9 3, 63 ÷ 9 7, 72 ÷ 9 8 Now, 3, 7, and 8 have no common factors.

The HCF is the last non-zero remainder, which is 2 × 9 18.