The least number that should be subtracted from the number 32146 to make it a perfect square is :

Options

A. 305
B. 105
C. 205
D. 405

Answer

To determine the least number that should be subtracted from 32146 to make it a perfect square, we can follow these steps:

### Step 1: Find the square root of 32146.
To find the closest perfect square, we first calculate the square root of 32146.

\[ \sqrt{32146} \approx 179.3 \]

### Step 2: Identify the nearest whole numbers.
The nearest whole numbers to 179.3 are 179 and 180. We will calculate the squares of both numbers.

\[ 179^2 = 32041 \]
\[ 180^2 = 32400 \]

### Step 3: Determine the perfect squares.
Now we have two perfect squares:
- \(179^2 = 32041\)
- \(180^2 = 32400\)

### Step 4: Calculate the differences.
Next, we find the differences between 32146 and these perfect squares.

1. Difference with \(179^2\):
\[ 32146 - 32041 = 105 \]

2. Difference with \(180^2\):
\[ 32400 - 32146 = 254 \]

### Step 5: Identify the least number to subtract.
The least number that needs to be subtracted from 32146 to make it a perfect square is the smaller of the two differences calculated.

\[ \text{Least number to subtract} = 105 \]

Thus, the least number that should be subtracted from 32146 to make it a perfect square is **105**.

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