Which smallest number must be added to 2203 so that we get a perfect square ?

Options

A. 1
B. 3
C. 6
D. 8

Answer

To find the smallest number that must be added to 2203 to make it a perfect square, we can follow these steps:

### Step 1: Identify the nearest perfect squares
We need to find the perfect squares that are close to 2203. The perfect squares are obtained by squaring whole numbers. We can start by estimating the square root of 2203.

### Step 2: Estimate the square root
Calculating the square root of 2203 gives us approximately 47 (since \(47^2 = 2209\) and \(46^2 = 2116\)). This means that the perfect square we are looking for is likely around 47.

### Step 3: Calculate the perfect squares
- \(46^2 = 2116\)
- \(47^2 = 2209\)
- \(48^2 = 2304\)

From this, we can see that:
- The nearest perfect square less than 2203 is \(46^2 = 2116\).
- The nearest perfect square greater than 2203 is \(47^2 = 2209\).

### Step 4: Determine the difference
Now, we need to find the difference between 2203 and the next perfect square, which is \(2209\):
\[ 2209 - 2203 = 6 \]

### Step 5: Conclusion
The smallest number that must be added to 2203 to make it a perfect square is **6**.

### Summary of the Steps:
1. Estimate the square root of 2203.
2. Identify the nearest whole numbers whose squares are close to 2203.
3. Calculate the squares of those numbers.
4. Find the difference between 2203 and the nearest perfect square greater than it.
5. Conclude that the smallest number to be added is 6.

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