Find the smallest number by which `10368` be divided, so that the result is a perfect square. Also, find the square root of the resulting number.

Answer

To solve the problem of finding the smallest number by which `10368` must be divided to make it a perfect square, and then finding the square root of the resulting number, we can follow these steps:

### Step 1: Prime Factorization of 10368
We start by finding the prime factors of `10368`.

1. Divide `10368` by `2` (the smallest prime number):
- \( 10368 \div 2 = 5184 \)
2. Divide `5184` by `2`:
- \( 5184 \div 2 = 2592 \)
3. Divide `2592` by `2`:
- \( 2592 \div 2 = 1296 \)
4. Divide `1296` by `2`:
- \( 1296 \div 2 = 648 \)
5. Divide `648` by `2`:
- \( 648 \div 2 = 324 \)
6. Divide `324` by `2`:
- \( 324 \div 2 = 162 \)
7. Divide `162` by `2`:
- \( 162 \div 2 = 81 \)

Now, `81` is not divisible by `2`, so we move to the next prime number, which is `3`.

8. Divide `81` by `3`:
- \( 81 \div 3 = 27 \)
9. Divide `27` by `3`:
- \( 27 \div 3 = 9 \)
10. Divide `9` by `3`:
- \( 9 \div 3 = 3 \)
11. Divide `3` by `3`:
- \( 3 \div 3 = 1 \)

The prime factorization of `10368` is:
\[ 10368 = 2^7 \times 3^4 \]

### Step 2: Determine the Exponents for Perfect Square
For a number to be a perfect square, all the exponents in its prime factorization must be even.

- The exponent of `2` is `7` (which is odd).
- The exponent of `3` is `4` (which is even).

To make the exponent of `2` even, we need to remove one `2`. Therefore, we divide `10368` by `2`:

### Step 3: Calculate the Resulting Number
Now, we divide `10368` by `2`:
\[ 10368 \div 2 = 5184 \]

### Step 4: Verify that 5184 is a Perfect Square
Now we check the prime factorization of `5184`:
\[ 5184 = 2^6 \times 3^4 \]
Both exponents are even, confirming that `5184` is a perfect square.

### Step 5: Find the Square Root of 5184
To find the square root of `5184`, we take the square root of each prime factor:
\[ \sqrt{5184} = \sqrt{2^6} \times \sqrt{3^4} = 2^{6/2} \times 3^{4/2} = 2^3 \times 3^2 = 8 \times 9 = 72 \]

### Final Answers
- The smallest number by which `10368` must be divided to make it a perfect square is `2`.
- The square root of the resulting number `5184` is `72`.