Finding the HCF of 36, 54, and 108

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. To find the HCF of 36, 54, and 108, we can use the prime factorization method.

Prime Factorization Method Explained

This method involves the following steps:

1. Find the prime factorization of each number.
2. Identify the common prime factors among all the numbers.
3. For each common prime factor, take the lowest power that appears in any of the factorizations.
4. Multiply these lowest powers of common prime factors together to get the HCF.

Applying Prime Factorization to 36, 54, and 108

Let's find the prime factors for each number:

• Prime factorization of 36: $36 = 2 \times 18 = 2 \times 2 \times 9 = 2^2 \times 3^2$
• Prime factorization of 54: $54 = 2 \times 27 = 2 \times 3 \times 9 = 2 \times 3^3$
• Prime factorization of 108: $108 = 2 \times 54 = 2 \times 2 \times 27 = 2^2 \times 3^3$

Now, let's list the prime factorizations:

• $36 = 2^2 \times 3^2$
• $54 = 2^1 \times 3^3$
• $108 = 2^2 \times 3^3$

Identify the common prime factors. Both 2 and 3 are common prime factors of 36, 54, and 108.

For each common prime factor, take the lowest power:

• For the prime factor 2, the powers are $2^2$ (from 36), $2^1$ (from 54), and $2^2$ (from 108). The lowest power is $2^1$.
• For the prime factor 3, the powers are $3^2$ (from 36), $3^3$ (from 54), and $3^3$ (from 108). The lowest power is $3^2$.

Multiply these lowest powers together to find the HCF:

$\text{HCF}(36, 54, 108) = 2^1 \times 3^2 = 2 \times (3 \times 3) = 2 \times 9 = 18$

Thus, the HCF of 36, 54, and 108 is 18.

We can also list the factors of each number to verify:

• Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
• Factors of 54: 1, 2, 3, 6, 9, 18, 27, 54
• Factors of 108: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108

The common factors are 1, 2, 3, 6, 9, and 18. The highest among these common factors is 18. This confirms our result.

Conclusion on the HCF

Based on the prime factorization and listing of factors, the Highest Common Factor of 36, 54, and 108 is indeed 18.

Revision Table: Understanding HCF

Additional Information: Related Concepts in Number Theory

Understanding HCF is fundamental in number theory. Here are some related concepts:

• Least Common Multiple (LCM): The smallest positive integer that is a multiple of two or more numbers. HCF and LCM are related by the formula for two numbers 'a' and 'b': $\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b$. This formula can be extended for three or more numbers, but the relationship is more complex.
• Co-prime Numbers: Two numbers are co-prime (or relatively prime) if their HCF is 1. For example, 7 and 10 are co-prime because $\text{HCF}(7, 10) = 1$.
• Euclidean Algorithm: An efficient method for computing the HCF of two integers. It involves repeatedly applying the division algorithm until the remainder is 0. The last non-zero remainder is the HCF. For finding HCF of three numbers, you can find HCF of the first two numbers, and then find the HCF of the result and the third number.

These concepts are interconnected and help in solving various problems involving divisibility and number properties.