Find the least number which must be subtracted from `2037` so that the resulting number is a perfect square.

Answer

To find the least number that must be subtracted from 2037 to make it a perfect square, we can follow these steps:

### Step 1: Identify the number
We start with the number 2037.

### Step 2: Use the long division method to find the nearest perfect square
We will apply the long division method to find the largest perfect square less than or equal to 2037.

1. **Pair the digits**: Starting from the right, we pair the digits of 2037. This gives us the pairs: 20 and 37.

2. **Find the largest square less than the first pair (20)**: The largest perfect square less than 20 is \(4^2 = 16\). So, we take 4 as our divisor.

3. **Subtract**:
\[ 20 - 16 = 4 \]
Here, 4 is the remainder.

4. **Bring down the next pair (37)**: We now have 437.

### Step 3: Continue the long division
1. **Double the divisor**: We double the number we used (4) to get 8.

2. **Find the next digit**: We need to find a digit \(x\) such that \( (80 + x) \times x \) is less than or equal to 437.

- Testing \(x = 5\):
\[ (80 + 5) \times 5 = 85 \times 5 = 425 \]
This is less than 437.

3. **Subtract**:
\[ 437 - 425 = 12 \]

4. **Double the new divisor**: Now we have 90 (which is \(80 + 10\)).

### Step 4: Check for the next digit
1. **Find the next digit**: We need to find \(y\) such that \( (90 + y) \times y \) is less than or equal to 12.

- Since \(90\) is already greater than \(12\), we cannot find a suitable \(y\) that satisfies this condition.

### Step 5: Conclusion
Since we have a remainder of 12, we conclude that 2037 is not a perfect square. To make it a perfect square, we need to subtract the remainder from 2037.

\[ 2037 - 12 = 2025 \]

### Step 6: Verify
We check if 2025 is a perfect square:
\[ \sqrt{2025} = 45 \]
Thus, 2025 is indeed a perfect square.

### Final Answer
The least number that must be subtracted from 2037 to make it a perfect square is **12**.

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