What is the Highest Common Factor of `2^(3) xx 3^(5)` and `3^(3) xx 5^(2)` ?

Options

A. `2^(3) xx 3^(5) xx 5^(2)`
B. `3^(3)`
C. `2^(1) xx 3^(3) xx 5^(1)`
D. `3^(5) xx 5^(2)`

Answer

To find the Highest Common Factor (HCF) of the numbers \(2^3 \times 3^5\) and \(3^3 \times 5^2\), we will follow these steps:

### Step 1: Identify the prime factorization of both terms
- The first term is \(2^3 \times 3^5\).
- The second term is \(3^3 \times 5^2\).

### Step 2: List the prime factors
- For the first term \(2^3 \times 3^5\), the prime factors are:
- \(2\) with exponent \(3\)
- \(3\) with exponent \(5\)

- For the second term \(3^3 \times 5^2\), the prime factors are:
- \(3\) with exponent \(3\)
- \(5\) with exponent \(2\)

### Step 3: Determine the common prime factors
- The common prime factor between both terms is \(3\).
- The other prime factors \(2\) and \(5\) are not common in both terms.

### Step 4: Find the minimum exponent for the common prime factor
- For the prime factor \(3\):
- In the first term, the exponent is \(5\).
- In the second term, the exponent is \(3\).
- The minimum exponent of \(3\) is \(3\).

### Step 5: Write the HCF
- Since \(2\) is not a common factor and \(5\) is not present in the first term, the HCF is:
\[ HCF = 3^3 \]

### Final Answer
The Highest Common Factor of \(2^3 \times 3^5\) and \(3^3 \times 5^2\) is \(3^3\).

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