Find the smallest number by which 2925 muse be divided to obtain a perfect square. Also find the square root of the perfect square so obtained.

Answer

To find the smallest number by which 2925 must be divided to obtain a perfect square, we will follow these steps:

### Step 1: Prime Factorization of 2925
First, we need to find the prime factorization of 2925.

- Divide 2925 by 5 (the smallest prime number):
\[ 2925 \div 5 = 585 \]
- Divide 585 by 5:
\[ 585 \div 5 = 117 \]
- Divide 117 by 3:
\[ 117 \div 3 = 39 \]
- Divide 39 by 3:
\[ 39 \div 3 = 13 \]
- 13 is a prime number.

So, the prime factorization of 2925 is:
\[ 2925 = 5^2 \times 3^2 \times 13^1 \]

### Step 2: Identify the Exponents
Next, we look at the exponents of the prime factors in the factorization:
- For \(5^2\), the exponent is 2 (even).
- For \(3^2\), the exponent is 2 (even).
- For \(13^1\), the exponent is 1 (odd).

### Step 3: Make Exponents Even
To make the number a perfect square, all prime factors must have even exponents. The only odd exponent we have is for \(13\).

To make it even, we need to multiply by \(13^1\) (since \(1 + 1 = 2\) which is even). Therefore, we need to divide by \(13\) to remove the odd exponent.

### Step 4: Calculate the Smallest Number
Thus, the smallest number by which 2925 must be divided to obtain a perfect square is:
\[ \text{Smallest number} = 13 \]

### Step 5: Find the Perfect Square
Now, we will divide 2925 by 13 to find the perfect square:
\[ \frac{2925}{13} = 225 \]

### Step 6: Find the Square Root
Finally, we find the square root of the perfect square \(225\):
\[ \sqrt{225} = 15 \]

### Conclusion
The smallest number by which 2925 must be divided to obtain a perfect square is \(13\), and the square root of the perfect square obtained is \(15\).

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