Find the HCF of `2 xx 3 ^(3) xx 5 ^(2) xx 7 ^(2) and 3 xx 5 ^(2) xx 7 xx 11 ^(2).`

Options

A. `3 xx 5 xx 7`
B. `2 xx 3 xx 5 xx 7 xx 11`
C. `3 xx 5 ^(2) xx 7`
D. `3 ^(3) xx 5 ^(2) xx 7 ^(2)`

Answer

To find the HCF (Highest Common Factor) of the two numbers given in their prime factorized form, we will follow these steps:

### Step 1: Write down the prime factorization of both numbers.
The two numbers are:
1. \( 2 \times 3^3 \times 5^2 \times 7^2 \)
2. \( 3^1 \times 5^2 \times 7^1 \times 11^2 \)

### Step 2: Identify the common prime factors.
Now we will identify the prime factors present in both numbers:
- For the first number: \( 2, 3, 5, 7 \)
- For the second number: \( 3, 5, 7, 11 \)

### Step 3: Determine the lowest power of each common prime factor.
- **For 2**: Present in the first number (power = 1), not present in the second number (power = 0). Hence, it is not common.
- **For 3**: Present in the first number (power = 3) and in the second number (power = 1). The lowest power is \( 3^1 \).
- **For 5**: Present in both numbers (power = 2 in both). The lowest power is \( 5^2 \).
- **For 7**: Present in the first number (power = 2) and in the second number (power = 1). The lowest power is \( 7^1 \).
- **For 11**: Present only in the second number (power = 2), hence it is not common.

### Step 4: Write the HCF using the common prime factors.
Now, we can write the HCF by multiplying the common prime factors raised to their lowest powers:
\[ HCF = 3^1 \times 5^2 \times 7^1 \]

### Step 5: Calculate the HCF.
Calculating the values:
\[ HCF = 3 \times 25 \times 7 \]
\[ = 3 \times 25 = 75 \]
\[ 75 \times 7 = 525 \]

### Final Answer:
The HCF of \( 2 \times 3^3 \times 5^2 \times 7^2 \) and \( 3 \times 5^2 \times 7 \times 11^2 \) is **525**.

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