To solve this problem, we need to determine the smallest number by which 2352 should be multiplied to make the resulting product a perfect square.

Steps:

1. Prime Factorization of 2352: $2352 \div 2 = 1176$ $1176 \div 2 = 588$ $588 \div 2 = 294$ $294 \div 2 = 147$ $147 \div 3 = 49$ $49 \div 7 = 7$ $7 \div 7 = 1$ So, the prime factorization of 2352 is: $2352 = 2^4 \times 3 \times 7^2$

2. Check for pairs in the prime factorization:

   • $2^4$ (has pairs of 2's, so it is already a perfect square component)
   • $3$ (no pair, needs one more 3 to make a pair)
   • $7^2$ (already a pair)

3. Determine the smallest multiplier:

   • To make 2352 a perfect square, we need to multiply it by 3, to get another factor of 3.

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

Verification:

Multiply 2352 by 3 and check if it becomes a perfect square: $2352 \times 3 = 7056$ The prime factorization of 7056 is: $7056 = 2^4 \times 3^2 \times 7^2$ Since all prime factors are now paired, 7056 is indeed a perfect square.

Thus, 3 is the correct answer.

Would you like further details on this solution? Here are some related questions you might consider:

1. What is a perfect square, and how is it determined?
2. How does prime factorization help in identifying perfect squares?
3. Can you explain the concept of least common multiple (LCM) in similar contexts?
4. What if the number was 2400 instead of 2352? What would the result be?
5. How does the process change if the number has multiple unpaired prime factors?

Tip: When solving problems involving perfect squares, always check for the pairing of prime factors.