Understanding 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 each of the numbers without leaving a remainder. When numbers are given in their prime factorization form, finding the HCF is straightforward.

Steps to Find HCF using Prime Factorization

To find the HCF of two numbers expressed as a product of prime powers:

1. Identify all the prime factors that are common to both numbers.
2. For each common prime factor, take the lowest power (exponent) of that factor that appears in either factorization.
3. Multiply these lowest powers of the common prime factors together. This product is the HCF.

Applying the Method to the Given Numbers

We are given two numbers:

• Number 1: $2^3 \times 3^5$
• Number 2: $3^3 \times 5^2$

Let's identify the prime factors in each number and their powers:

Now, let's find the common prime factors and their lowest powers:

• The prime factor 2 is present in Number 1 (power 3) but not in Number 2. It is not a common factor.
• The prime factor 3 is present in Number 1 (power 5) and in Number 2 (power 3). It is a common factor. The lowest power of 3 is 3.
• The prime factor 5 is present in Number 2 (power 2) but not in Number 1. It is not a common factor.

The only common prime factor is 3, and its lowest power is 3. Therefore, the HCF is $3^3$.

Calculation of the HCF

Based on the analysis, the HCF is the product of the common prime factors raised to their lowest powers.

Common prime factor: 3

Lowest power of 3: 3

HCF = $3^3$

Conclusion on the Highest Common Factor

The Highest Common Factor of $2^3 \times 3^5$ and $3^3 \times 5^2$ is $3^3$. This means $3^3$ is the largest number that can divide both $2^3 \times 3^5$ and $3^3 \times 5^2$ completely without leaving a remainder.

Revision Table: HCF Prime Factorization

Additional Information on HCF and LCM

While HCF uses the lowest powers of common prime factors, the Least Common Multiple (LCM) uses the highest powers of all prime factors present in either number. For the same numbers, $2^3 \times 3^5$ and $3^3 \times 5^2$:

• Prime factors involved: 2, 3, 5.
• Highest power of 2: $2^3$ (from the first number).
• Highest power of 3: $3^5$ (from the first number).
• Highest power of 5: $5^2$ (from the second number).

So, the LCM would be $2^3 \times 3^5 \times 5^2$. Understanding both HCF and LCM helps in solving various problems involving fractions, ratios, and number theory.