Q#4: Is 2352 a perfect square? if not, find the smallest number by which 

2352 must be multiplied so that the product is a perfect.

To determine if 2352 is a perfect square, we can perform a prime factorization of the number.

### Step 1: Prime Factorization
Let's find the prime factors of 2352:

1. 2352 is even, so we start with dividing by 2:
   \[
   2352 \div 2 = 1176
   \]
2. 1176 is also even:
   \[
   1176 \div 2 = 588
   \]
3. 588 is also even:
   \[
   588 \div 2 = 294
   \]
4. 294 is also even:
   \[
   294 \div 2 = 147
   \]
5. 147 is not even but divisible by 3 (since the sum of the digits, 1+4+7=12, is divisible by 3):
   \[
   147 \div 3 = 49
   \]
6. 49 is \(7^2\) (since \(49 = 7 \times 7\)).

So, the prime factorization of 2352 is:
\[
2352 = 2^4 \times 3 \times 7^2
\]

### Step 2: Check if the Number is a Perfect Square
For a number to be a perfect square, all the exponents in its prime factorization must be even.

In the factorization \(2^4 \times 3 \times 7^2\):
- The exponent of 2 is 4 (even)
- The exponent of 3 is 1 (odd)
- The exponent of 7 is 2 (even)

Since the exponent of 3 is odd, 2352 is not a perfect square.

### Step 3: Find the Smallest Multiplier to Make it a Perfect Square
To make the product a perfect square, we need to make all the exponents even. The only exponent that is currently odd is 1 (for the prime number 3). To make it even, we need one more 3 (since \(3^1 \times 3^1 = 3^2\)).

So, we must multiply 2352 by 3 to make it a perfect square.

### Calculation:
\[
2352 \times 3 = 7056
\]

### Verification:
We can verify by finding the square root of 7056 to confirm it is an integer:
\[
\sqrt{7056} = 84
\]
Since 84 is an integer, 7056 is indeed a perfect square.

Therefore, the smallest number by which 2352 must be multiplied to make it a perfect square is **3**.

Would you like more details or have any questions about this solution?

Here are 8 related questions to expand your understanding:
1. How do you determine if a number is a perfect square?
2. What are the steps involved in prime factorization?
3. Why must all exponents in the prime factorization be even for a number to be a perfect square?
4. Can you provide an example of another number and determine if it is a perfect square?
5. How can you find the prime factors of a large number efficiently?
6. What is the significance of multiplying by the smallest number to achieve a perfect square?
7. How do you verify if a product is a perfect square using its square root?
8. What are some practical applications of identifying perfect squares?

**Tip:** When performing prime factorization, always start with the smallest prime number and proceed to larger ones until the number is completely factored.

Explore if 2352 is a perfect square and discover the smallest number by which it must be multiplied to become a perfect square. Includes prime factorization steps, verification of perfect square, and practical applications.

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