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 first need to express both numbers in their prime factorization form.

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

### Step 1: Identify the prime factors of \( P \) and \( Q \)

- For \( P \):
- Prime factors: \( 2, 3, 5 \)
- Exponents: \( 3, 10, 1 \)

- For \( Q \):
- Prime factors: \( 2, 3, 7 \)
- Exponents: \( 5, 1, 1 \)

### Step 2: Find the minimum exponent for each common prime factor

Now, we will find the HCF by taking the lowest power of each common prime factor:

- For the prime factor \( 2 \):
- In \( P \), the exponent is \( 3 \)
- In \( Q \), the exponent is \( 5 \)
- Minimum exponent = \( \min(3, 5) = 3 \)

- For the prime factor \( 3 \):
- In \( P \), the exponent is \( 10 \)
- In \( Q \), the exponent is \( 1 \)
- Minimum exponent = \( \min(10, 1) = 1 \)

- For the prime factor \( 5 \):
- In \( P \), the exponent is \( 1 \)
- In \( Q \), the exponent is \( 0 \) (since \( 5 \) is not a factor of \( Q \))
- Minimum exponent = \( \min(1, 0) = 0 \)

- For the prime factor \( 7 \):
- In \( P \), the exponent is \( 0 \) (since \( 7 \) is not a factor of \( P \))
- In \( Q \), the exponent is \( 1 \)
- Minimum exponent = \( \min(0, 1) = 0 \)

### Step 3: Write the HCF using the minimum exponents

Now we can write the HCF using the common prime factors and their minimum exponents:

\[ \text{HCF}(P, Q) = 2^{3} \cdot 3^{1} \cdot 5^{0} \cdot 7^{0} \]

Since \( 5^{0} = 1 \) and \( 7^{0} = 1 \), we can simplify this to:

\[ \text{HCF}(P, Q) = 2^{3} \cdot 3^{1} = 8 \cdot 3 = 24 \]

### Final Answer:
The HCF of \( P \) and \( Q \) is \( 24 \).

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