By what least number should `6300` be divided to get a perfect square number ? Find the number whose square is the new number.

Answer

To solve the problem, we need to find the least number by which `6300` should be divided to obtain a perfect square number. We will also find the square root of the resulting perfect square.

### Step-by-Step Solution:

**Step 1: Prime Factorization of 6300**
We start by performing the prime factorization of `6300`.

- Divide by `2`:
\( 6300 \div 2 = 3150 \)
\( 3150 \div 2 = 1575 \) (2 appears twice)

- Divide by `3`:
\( 1575 \div 3 = 525 \)
\( 525 \div 3 = 175 \) (3 appears twice)

- Divide by `5`:
\( 175 \div 5 = 35 \)
\( 35 \div 5 = 7 \) (5 appears twice)

- Finally, we have `7` which is a prime number.

So, the prime factorization of `6300` is:
\[ 6300 = 2^2 \times 3^2 \times 5^2 \times 7^1 \]

**Step 2: Identify the Exponents**
Next, we look at the exponents of the prime factors:
- \( 2^2 \) (even)
- \( 3^2 \) (even)
- \( 5^2 \) (even)
- \( 7^1 \) (odd)

For a number to be a perfect square, all the exponents in its prime factorization must be even.

**Step 3: Determine the Least Number to Divide**
Since `7` has an odd exponent, we need to divide `6300` by `7` to make the exponent even.

**Step 4: Calculate the Resulting Number**
Now, we divide `6300` by `7`:
\[ 6300 \div 7 = 900 \]

**Step 5: Verify if 900 is a Perfect Square**
Next, we check if `900` is a perfect square by finding its square root:
\[ \sqrt{900} = 30 \]

Thus, `900` is indeed a perfect square.

### Final Answer:
The least number by which `6300` should be divided to get a perfect square is `7`, and the number whose square is the new number is `30`.