(i) Find the smallest number by which `2592` be multiplied so that the product is a perfect square.
(ii) Find the smallest number by which `12748` be multiplied so that the product is a perfect square.

Answer

To solve the problem, we will find the smallest number by which `2592` and `12748` need to be multiplied to make them perfect squares. We will do this by finding their prime factors and determining what is needed to form pairs.

### Part (i): Finding the smallest number for `2592`

1. **Find the prime factorization of 2592:**
- Divide by 2:
- \( 2592 \div 2 = 1296 \)
- \( 1296 \div 2 = 648 \)
- \( 648 \div 2 = 324 \)
- \( 324 \div 2 = 162 \)
- \( 162 \div 2 = 81 \) (81 is not divisible by 2)
- Now divide by 3:
- \( 81 \div 3 = 27 \)
- \( 27 \div 3 = 9 \)
- \( 9 \div 3 = 3 \)
- \( 3 \div 3 = 1 \)
- The prime factorization of `2592` is:
\[ 2592 = 2^5 \times 3^4 \]

2. **Identify the pairs of prime factors:**
- For \( 2^5 \): There are 5 twos, which gives us 2 pairs and 1 leftover.
- For \( 3^4 \): There are 4 threes, which gives us 2 pairs.

3. **Determine the missing pairs:**
- The factor \( 2^5 \) has 1 unpaired 2.
- The factor \( 3^4 \) has all pairs.

4. **Multiply by the smallest number to make pairs:**
- To make a pair for the unpaired 2, we need to multiply by \( 2 \).
- Therefore, the smallest number to multiply `2592` by is \( 2 \).

5. **Final result for part (i):**
- The smallest number is \( 2 \).

### Part (ii): Finding the smallest number for `12748`

1. **Find the prime factorization of 12748:**
- Divide by 2:
- \( 12748 \div 2 = 6374 \)
- \( 6374 \div 2 = 3187 \) (3187 is not divisible by 2)
- Now check if 3187 is a prime number:
- It is not divisible by any prime numbers up to its square root, so it remains as \( 3187 \).
- The prime factorization of `12748` is:
\[ 12748 = 2^2 \times 3187^1 \]

2. **Identify the pairs of prime factors:**
- For \( 2^2 \): There are 2 twos, which gives us 1 pair.
- For \( 3187^1 \): There is 1 unpaired 3187.

3. **Determine the missing pairs:**
- The factor \( 2^2 \) has all pairs.
- The factor \( 3187^1 \) has 1 unpaired.

4. **Multiply by the smallest number to make pairs:**
- To make a pair for the unpaired 3187, we need to multiply by \( 3187 \).
- Therefore, the smallest number to multiply `12748` by is \( 3187 \).

5. **Final result for part (ii):**
- The smallest number is \( 3187 \).

### Summary of Results:
- (i) The smallest number to multiply `2592` by is **2**.
- (ii) The smallest number to multiply `12748` by is **3187**.