What is the least number which must be subtracted from 10420 to make it a perfect square?

Options

A. 219
B. 200
C. 189
D. 16

Answer

To find the least number that must be subtracted from 10420 to make it a perfect square, we can follow these steps:

### Step 1: Find the square root of 10420
First, we need to estimate the square root of 10420. The perfect squares close to 10420 are \(100^2 = 10000\) and \(101^2 = 10201\).

### Step 2: Determine the nearest perfect square
Since \(100^2 = 10000\) and \(101^2 = 10201\), we can see that 10420 is between these two perfect squares. The nearest perfect square less than 10420 is \(100^2 = 10000\).

### Step 3: Calculate the difference
Now, we subtract the nearest perfect square from 10420:
\[ 10420 - 10000 = 420 \]

### Step 4: Check the next perfect square
Next, we check the next perfect square, which is \(101^2 = 10201\):
\[ 10420 - 10201 = 219 \]

### Step 5: Identify the least number to subtract
Now we compare the two differences we found:
- To make 10420 a perfect square of \(100\), we need to subtract \(420\).
- To make it a perfect square of \(101\), we need to subtract \(219\).

Since \(219 < 420\), the least number that must be subtracted from 10420 to make it a perfect square is:
\[ \boxed{219} \]