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Calculate the HCF of p^(3)q^(2) and p^(2...

Calculate the HCF of `p^(3)q^(2) and p^(2)q,` provided that p and q are prime numbers ?

A

pq

B

`pq^(2)`

C

`p^(2)q`

D

`p^(2)q^(2)`

Text Solution

AI Generated Solution

The correct Answer is:
To calculate the HCF (Highest Common Factor) of \( p^3q^2 \) and \( p^2q \), where \( p \) and \( q \) are prime numbers, we can follow these steps: ### Step-by-Step Solution: 1. **Identify the Expressions**: We have two expressions: - \( M = p^3 q^2 \) - \( N = p^2 q \) 2. **Factor Each Expression**: - For \( M = p^3 q^2 \), we can express it as: \[ M = p \cdot p \cdot p \cdot q \cdot q \] - For \( N = p^2 q \), we can express it as: \[ N = p \cdot p \cdot q \] 3. **Identify Common Factors**: - The common prime factor in both expressions is \( p \) and \( q \). - For \( p \): - In \( M \), \( p \) appears 3 times (i.e., \( p^3 \)). - In \( N \), \( p \) appears 2 times (i.e., \( p^2 \)). - For \( q \): - In \( M \), \( q \) appears 2 times (i.e., \( q^2 \)). - In \( N \), \( q \) appears 1 time (i.e., \( q \)). 4. **Determine the HCF for Each Factor**: - For \( p \), the minimum power is \( \min(3, 2) = 2 \), so we take \( p^2 \). - For \( q \), the minimum power is \( \min(2, 1) = 1 \), so we take \( q^1 \) or simply \( q \). 5. **Combine the Common Factors**: - Thus, the HCF is: \[ \text{HCF} = p^2 q \] ### Final Answer: The HCF of \( p^3 q^2 \) and \( p^2 q \) is \( p^2 q \). ---
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