The smallest number must be added to 1780 to make it a perfect square is :

Options

A. 69
B. 156
C. 49
D. 59

Answer

To find the smallest number that must be added to 1780 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 1780. The perfect squares are numbers like \(1^2, 2^2, 3^2, \ldots\).

### Step 2: Calculate the square root of 1780
First, we calculate the approximate square root of 1780.
\[ \sqrt{1780} \approx 42.1 \]
This means the perfect squares we are interested in are \(42^2\) and \(43^2\).

### Step 3: Calculate \(42^2\) and \(43^2\)
Now we calculate these squares:
\[ 42^2 = 1764 \]
\[ 43^2 = 1849 \]

### Step 4: Determine the difference between 1780 and the nearest perfect square
Now we find the difference between 1780 and the nearest perfect square below it, which is \(1764\):
\[ 1780 - 1764 = 16 \]
And the difference between 1780 and the next perfect square above it, which is \(1849\):
\[ 1849 - 1780 = 69 \]

### Step 5: Identify the smallest number to be added
Since we are looking for the smallest number to be added to 1780 to make it a perfect square, we compare the two differences:
- To reach \(1764\), we would have to subtract \(16\) (not applicable since we want to add).
- To reach \(1849\), we need to add \(69\).

Thus, the smallest number that must be added to 1780 to make it a perfect square is:
\[ \text{Smallest number to add} = 69 \]

### Final Answer
The smallest number that must be added to 1780 to make it a perfect square is **69**.

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