What least number must be substracted from 176 to make it a perfect square?

Options

A. 16
B. 10
C. 7
D. 4

Answer

To find the least number that must be subtracted from 176 to make it a perfect square, we can follow these steps:

### Step 1: Identify if 176 is a perfect square
First, we need to check if 176 is a perfect square. A perfect square is a number that can be expressed as the square of an integer.

### Step 2: Calculate the square root of 176
To find the nearest perfect square, we can calculate the square root of 176 using the long division method.

1. **Pair the digits**: Start from the right. Pair the digits of 176, which gives us (1)(76).
2. **Find the largest square**: The largest square less than or equal to 1 is 1 (since \(1^2 = 1\)).
3. **Subtract**: Subtract 1 from 1, which gives us 0. Bring down the next pair (76) to get 76.
4. **Double the root**: The current root is 1, double it to get 2.
5. **Find the next digit**: We need to find a digit \(x\) such that \(2x \cdot x \leq 76\). Testing \(x = 3\):
- \(23 \cdot 3 = 69\) (which is less than 76).
6. **Subtract**: Subtract 69 from 76, giving us a remainder of 7.

### Step 3: Determine the least number to subtract
Since the remainder is 7, this means that if we subtract 7 from 176, we will get a perfect square.

### Step 4: Calculate the resulting number
Now, subtract 7 from 176:
\[ 176 - 7 = 169 \]

### Step 5: Verify if 169 is a perfect square
Finally, check if 169 is a perfect square:
\[ \sqrt{169} = 13 \]
Since 13 is an integer, 169 is indeed a perfect square.

### Conclusion
The least number that must be subtracted from 176 to make it a perfect square is **7**.

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