If `52272 = p^(2) xx q^(3) xx r^(4)`, where p, q and r are prime numbers, then the value of `(2p + q - r)` is

Options

A. 29
B. 21
C. 22
D. 23

Answer

To solve the problem, we need to perform the prime factorization of the number 52272 and express it in the form \( p^2 \times q^3 \times r^4 \), where \( p, q, \) and \( r \) are prime numbers. Then we will calculate \( 2p + q - r \).

### Step-by-step Solution:

1. **Prime Factorization of 52272**:
- Start dividing 52272 by the smallest prime number, which is 2.
- \( 52272 \div 2 = 26136 \)
- \( 26136 \div 2 = 13068 \)
- \( 13068 \div 2 = 6534 \)
- \( 6534 \div 2 = 3267 \)
- Now, 3267 is not divisible by 2. Move to the next prime number, which is 3.
- \( 3267 \div 3 = 1089 \)
- \( 1089 \div 3 = 363 \)
- \( 363 \div 3 = 121 \)
- Now, 121 is not divisible by 3. Move to the next prime number, which is 11.
- \( 121 \div 11 = 11 \)
- \( 11 \div 11 = 1 \)

So, the prime factorization of 52272 is:
\[ 52272 = 2^4 \times 3^3 \times 11^2 \]

2. **Expressing in the required form**:
- We need to express \( 52272 \) in the form \( p^2 \times q^3 \times r^4 \).
- From our factorization, we can assign:
- \( p = 11 \) (since \( 11^2 \))
- \( q = 3 \) (since \( 3^3 \))
- \( r = 2 \) (since \( 2^4 \))

3. **Calculating \( 2p + q - r \)**:
- Substitute the values of \( p, q, \) and \( r \):
\[ 2p + q - r = 2(11) + 3 - 2 \]
- Calculate:
\[ = 22 + 3 - 2 \]
\[ = 23 \]

### Final Answer:
The value of \( 2p + q - r \) is \( 23 \).

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