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

Options

A. 2
B. 3
C. 4
D. 6

Answer

To find the smallest number such that multiplying it with 24 results in a perfect square, we can follow these steps:

### Step 1: Factor 24 into its prime factors
First, we need to factor the number 24 into its prime factors.

\[ 24 = 2^3 \times 3^1 \]

### Step 2: Identify the conditions for a perfect square
A number is a perfect square if all the exponents in its prime factorization are even. In our case, we have:

- The exponent of 2 is 3 (which is odd).
- The exponent of 3 is 1 (which is also odd).

### Step 3: Determine what is needed to make the exponents even
To make the exponents even, we need to adjust them:

- For \(2^3\), we need one more \(2\) to make it \(2^4\) (which is even).
- For \(3^1\), we need one more \(3\) to make it \(3^2\) (which is even).

### Step 4: Calculate the smallest number to multiply
The smallest number we need to multiply 24 by to achieve even exponents is:

\[ 2^1 \times 3^1 = 2 \times 3 = 6 \]

### Step 5: Verify the result
Now, we multiply 24 by 6:

\[ 24 \times 6 = 144 \]

Now, we check if 144 is a perfect square:

\[ 144 = 12^2 \]

Since 144 is indeed a perfect square, our solution is confirmed.

### Final Answer
The smallest number that needs to be multiplied with 24 to obtain a perfect square is **6**.

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