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

Question

Accepted Answer

To determine why Option 3 is the correct answer, we need to factorize the number 52272 into its prime factors and then use these factors to calculate the expression (2p + q – r).

### Step 1: Prime Factorization of 52272

First, let's factorize 52272:

1. **Divide by 2** (since 52272 is even):
   $$
   52272 \div 2 = 26136
   $$
2. **Divide by 2 again**:
   $$
   26136 \div 2 = 13068
   $$
3. **Divide by 2 again**:
   $$
   13068 \div 2 = 6534
   $$
4. **Divide by 2 again**:
   $$
   6534 \div 2 = 3267
   $$

Now, 3267 is not divisible by 2. Let's try the next smallest prime number, which is 3:

5. **Divide by 3**:
   $$
   3267 \div 3 = 1089
   $$
6. **Divide by 3 again**:
   $$
   1089 \div 3 = 363
   $$
7. **Divide by 3 again**:
   $$
   363 \div 3 = 121
   $$

Now, 121 is not divisible by 3. Let's try the next smallest prime number, which is 11:

8. **Divide by 11**:
   $$
   121 \div 11 = 11
   $$
9. **Divide by 11 again**:
   $$
   11 \div 11 = 1
   $$

Thus, the prime factorization of 52272 is:
$$ 
52272 = 2^4 \times 3^3 \times 11^2 
$$

### Step 2: Assign Values to p, q, and r

From the prime factorization, we can assign:
- $ p = 2 $
- $ q = 3 $
- $ r = 11 $

### Step 3: Calculate the Expression (2p + q – r)

Now, substitute the values of p, q, and r into the expression:
$$
2p + q - r = 2(2) + 3 - 11
$$
$$
= 4 + 3 - 11
$$
$$
= 7 - 11
$$
$$
= -4
$$

However, the correct answer is given as Option 3, which is 23. Let's re-evaluate the assignment of p, q, and r based on the powers in the factorization:

- Since $ p^2 $, $ q^3 $, and $ r^4 $ are the powers, we should have:
  - $ p = 11 $ (since $ 11^2 $)
  - $ q = 3 $ (since $ 3^3 $)
  - $ r = 2 $ (since $ 2^4 $)

Recalculate the expression with these assignments:
$$
2p + q - r = 2(11) + 3 - 2
$$
$$
= 22 + 3 - 2
$$
$$
= 23
$$

Thus, the correct value of the expression is indeed 23, which matches Option 3.

### Explanation of Incorrect Options

- **Option 1 (21)**: This would require a different combination of p, q, and r that does not fit the prime factorization.
- **Option 2 (22)**: This is close but incorrect due to the specific values of p, q, and r derived from the factorization.
- **Option 4 (29)**: This value is too high and cannot be achieved with the given prime factors.

Therefore, Option 3 is the correct answer, as it accurately reflects the calculation based on the correct assignment of prime factors.