The least possible divisor of 25930800 by which we divide this number, we get the quotient as a perfect square :

Options

A. 2
B. 3
C. 5
D. Can't be determined

Answer

To find the least possible divisor of 25930800 such that the quotient is a perfect square, we can follow these steps:

### Step 1: Understand the Problem
We need to find a divisor \( d \) of 25930800 such that when we divide 25930800 by \( d \), the result is a perfect square.

### Step 2: Factorization of the Number
First, we should factor the number 25930800 to understand its prime factors.

### Step 3: Prime Factorization
Let's perform the prime factorization of 25930800:
- Start dividing by the smallest prime number, which is 2.
- 25930800 ÷ 2 = 12965400
- 12965400 ÷ 2 = 6482700
- 6482700 ÷ 2 = 3241350
- 3241350 ÷ 2 = 1620675 (stop here as 1620675 is odd)

Now, divide by the next smallest prime number, which is 3:
- 1620675 ÷ 3 = 540225
- 540225 ÷ 3 = 180075
- 180075 ÷ 3 = 60025 (stop here as 60025 is not divisible by 3)

Next, divide by 5:
- 60025 ÷ 5 = 12005
- 12005 ÷ 5 = 2401 (stop here as 2401 is not divisible by 5)

Now, divide by 7:
- 2401 ÷ 7 = 343
- 343 ÷ 7 = 49
- 49 ÷ 7 = 7
- 7 ÷ 7 = 1

So, the complete prime factorization of 25930800 is:
\[ 25930800 = 2^4 \times 3^3 \times 5^2 \times 7^4 \]

### Step 4: Conditions for Perfect Square
For a number to be a perfect square, all the exponents in its prime factorization must be even.

### Step 5: Analyze the Exponents
From the factorization:
- \( 2^4 \) (even)
- \( 3^3 \) (odd)
- \( 5^2 \) (even)
- \( 7^4 \) (even)

### Step 6: Finding the Least Divisor
To make the quotient a perfect square, we need to make the exponent of 3 even. The least divisor we can use to achieve this is \( 3^1 \) (since \( 3^3 \) is odd, we need to divide by one \( 3 \) to make it \( 3^2 \), which is even).

### Step 7: Calculate the Quotient
Now, we divide 25930800 by 3:
\[ \frac{25930800}{3} = 8643600 \]

### Step 8: Check if the Quotient is a Perfect Square
Now we need to check if 8643600 is a perfect square:
- The prime factorization of 8643600 is \( 2^4 \times 3^2 \times 5^2 \times 7^4 \), which has all even exponents.

### Conclusion
Since all the exponents are even, 8643600 is indeed a perfect square.

Thus, the least possible divisor of 25930800 such that the quotient is a perfect square is:
\[ \boxed{3} \]