If `P=2^(3).3^(10).5: Q = 2^(5).3.7`, then HCF of P and Q is:

Options

A. `2.3.5.7`
B. `3.2^(3)`
C. `2^(2)3^(7)`
D. `2^(5).3^(10).5.7`

Answer

To find the HCF (Highest Common Factor) of the numbers \( P \) and \( Q \), we start by expressing both numbers in their prime factorization form.

### Step 1: Write down the prime factorization of \( P \) and \( Q \)

Given:
- \( P = 2^3 \times 3^{10} \times 5^1 \)
- \( Q = 2^5 \times 3^1 \times 7^1 \)

### Step 2: Identify the common prime factors

Next, we identify the common prime factors between \( P \) and \( Q \):
- The common prime factors are \( 2 \) and \( 3 \).
- The prime factor \( 5 \) in \( P \) and \( 7 \) in \( Q \) are not common.

### Step 3: Determine the minimum power of each common prime factor

Now, we take the minimum power for each of the common prime factors:
- For \( 2 \): The powers are \( 3 \) (from \( P \)) and \( 5 \) (from \( Q \)). The minimum is \( \min(3, 5) = 3 \).
- For \( 3 \): The powers are \( 10 \) (from \( P \)) and \( 1 \) (from \( Q \)). The minimum is \( \min(10, 1) = 1 \).

### Step 4: Write down the HCF using the minimum powers

Now we can write the HCF using the minimum powers we found:
\[ \text{HCF}(P, Q) = 2^{\min(3, 5)} \times 3^{\min(10, 1)} = 2^3 \times 3^1 \]

### Step 5: Calculate the HCF

Finally, we can compute the HCF:
\[ \text{HCF}(P, Q) = 2^3 \times 3^1 = 8 \times 3 = 24 \]

Thus, the HCF of \( P \) and \( Q \) is \( 24 \).