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 xx7 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 numbers \(2 \times 3^3 \times 5^2 \times 7^2\) and \(3 \times 5^2 \times 7 \times 11^2\), we can follow these steps:

### Step 1: Write down the prime factorization of both numbers.
- The first number is:
\[ 2^1 \times 3^3 \times 5^2 \times 7^2 \]
- The second number is:
\[ 3^1 \times 5^2 \times 7^1 \times 11^2 \]

### Step 2: Identify the common prime factors.
The common prime factors between the two numbers are \(3\), \(5\), and \(7\).

### Step 3: Determine the lowest power of each common prime factor.
- For \(3\):
- In the first number, the power is \(3\).
- In the second number, the power is \(1\).
- The lowest power is \(1\), so we take \(3^1\).

- For \(5\):
- In both numbers, the power is \(2\).
- The lowest power is \(2\), so we take \(5^2\).

- For \(7\):
- In the first number, the power is \(2\).
- In the second number, the power is \(1\).
- The lowest power is \(1\), so we take \(7^1\).

### Step 4: Multiply the common prime factors with their lowest powers to find the HCF.
Now we can calculate the HCF:
\[ \text{HCF} = 3^1 \times 5^2 \times 7^1 = 3 \times 25 \times 7 \]

### Step 5: Calculate the final value.
Calculating this gives:
\[ 3 \times 25 = 75 \]
\[ 75 \times 7 = 525 \]

Thus, the HCF of \(2 \times 3^3 \times 5^2 \times 7^2\) and \(3 \times 5^2 \times 7 \times 11^2\) is:
\[ \text{HCF} = 525 \]

### Final Answer:
The HCF is \(3 \times 5^2 \times 7 = 525\).

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