What will be the Highest Common Factor of `2^3xx3^5xx5^5 and 3^1xx5^2xx7^1`?

Options

A. `2^1xx3^1xx5^2`
B. `2^1xx3^1xx5^2xx7^1`
C. `2^3xx3^5xx5^7xx7^1`
D. `3^1xx5^2`

Answer

To find the Highest Common Factor (HCF) of the two numbers \(2^3 \times 3^5 \times 5^5\) and \(3^1 \times 5^2 \times 7^1\), we will follow these steps:

### Step 1: Identify the prime factors
The first number can be expressed as:
- \(2^3\)
- \(3^5\)
- \(5^5\)

The second number can be expressed as:
- \(3^1\)
- \(5^2\)
- \(7^1\)

### Step 2: List the prime factors with their powers
Now, we will list the prime factors and their corresponding powers from both numbers:

1. **For \(2^3 \times 3^5 \times 5^5\)**:
- \(2\) has a power of \(3\)
- \(3\) has a power of \(5\)
- \(5\) has a power of \(5\)

2. **For \(3^1 \times 5^2 \times 7^1\)**:
- \(3\) has a power of \(1\)
- \(5\) has a power of \(2\)
- \(7\) has a power of \(1\)

### Step 3: Determine the common prime factors
Next, we identify the common prime factors between the two numbers:
- The common prime factors are \(3\) and \(5\).

### Step 4: Take the minimum power of each common prime factor
Now, we take the minimum power of each common prime factor:
- For \(3\): The minimum power is \(\min(5, 1) = 1\)
- For \(5\): The minimum power is \(\min(5, 2) = 2\)

### Step 5: Write the HCF
Now we can write the HCF using the common prime factors and their minimum powers:
\[ \text{HCF} = 3^1 \times 5^2 \]

### Step 6: Calculate the HCF
Calculating the HCF:
\[ 3^1 = 3 \]
\[ 5^2 = 25 \]
\[ \text{HCF} = 3 \times 25 = 75 \]

### Final Answer
Thus, the Highest Common Factor (HCF) of \(2^3 \times 3^5 \times 5^5\) and \(3^1 \times 5^2 \times 7^1\) is \(75\).

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