Find the smallest number such that by multiplying it with 24, a complete square number is obtained :

Question

Accepted Answer

To determine the smallest number that, when multiplied by 24, results in a perfect square, we need to analyze the prime factorization of 24 and identify the additional factors required to form a perfect square.

1. **Prime Factorization of 24:**
   - 24 can be expressed as $24 = 2^3 \times 3^1$.

2. **Understanding Perfect Squares:**
   - A perfect square is a number that can be expressed as the square of an integer. For a number to be a perfect square, all the exponents in its prime factorization must be even.

3. **Analyzing the Prime Factors:**
   - In $24 = 2^3 \times 3^1$, the exponent of 2 is 3, and the exponent of 3 is 1. Both are odd numbers, which means 24 is not a perfect square.

4. **Adjusting the Exponents:**
   - To make the exponent of 2 even, we need to multiply by one more 2, changing $2^3$ to $2^4$.
   - To make the exponent of 3 even, we need to multiply by one more 3, changing $3^1$ to $3^2$.

5. **Calculating the Required Multiplier:**
   - Therefore, the smallest number we need to multiply by is $2 \times 3 = 6$.

6. **Verification:**
   - Multiplying 24 by 6 gives us $24 \times 6 = 144$.
   - The prime factorization of 144 is $2^4 \times 3^2$, which confirms that 144 is a perfect square ($12^2$).

**Conclusion:**
- Option 4 (6) is the correct answer because multiplying 24 by 6 results in a perfect square.

**Explanation of Incorrect Options:**
- **Option 1 (2):** Multiplying 24 by 2 gives $48 = 2^4 \times 3^1$, which is not a perfect square because the exponent of 3 is still odd.
- **Option 2 (3):** Multiplying 24 by 3 gives $72 = 2^3 \times 3^2$, which is not a perfect square because the exponent of 2 is still odd.
- **Option 3 (4):** Multiplying 24 by 4 gives $96 = 2^5 \times 3^1$, which is not a perfect square because both exponents are odd.

Thus, Option 4 (6) is the only choice that results in a perfect square when multiplied by 24.