Find the smallest number by which `3150` be divided, so that the quotient is a perfect square.

Answer

To find the smallest number by which 3150 must be divided so that the quotient is a perfect square, we can follow these steps:

### Step 1: Prime Factorization of 3150
We start by finding the prime factors of 3150.

1. Divide by the smallest prime number, 2:
\[ 3150 \div 2 = 1575 \]
2. Next, divide by 3:
\[ 1575 \div 3 = 525 \]
3. Divide by 3 again:
\[ 525 \div 3 = 175 \]
4. Now divide by 5:
\[ 175 \div 5 = 35 \]
5. Divide by 5 again:
\[ 35 \div 5 = 7 \]
6. Finally, divide by 7:
\[ 7 \div 7 = 1 \]

So, the prime factorization of 3150 is:
\[ 3150 = 2^1 \times 3^2 \times 5^2 \times 7^1 \]

### Step 2: Identify Unpaired Factors
In the prime factorization, we have:
- \(2^1\) (unpaired)
- \(3^2\) (paired)
- \(5^2\) (paired)
- \(7^1\) (unpaired)

To form a perfect square, all prime factors must have even exponents. Here, \(2\) and \(7\) have odd exponents.

### Step 3: Determine the Smallest Number to Divide
To make the exponents of all prime factors even, we need to pair the unpaired factors:
- For \(2^1\), we need one more \(2\) (to make it \(2^2\)).
- For \(7^1\), we need one more \(7\) (to make it \(7^2\)).

Thus, we need to multiply these unpaired factors:
\[ 2 \times 7 = 14 \]

### Step 4: Conclusion
Therefore, the smallest number by which 3150 must be divided to make the quotient a perfect square is:
\[ \boxed{14} \]

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