Find the square root of `276676`

Answer

To find the square root of \( 276676 \) using the long division method, we will follow these steps:

### Step 1: Pair the digits
Start from the right and pair the digits of the number. For \( 276676 \), we pair as follows:
- 27 | 66 | 76

### Step 2: Find the largest square
Look for the largest perfect square less than or equal to the first pair (27). The largest perfect square is \( 5^2 = 25 \).
- Write 5 above the line.

### Step 3: Subtract and bring down the next pair
Subtract \( 25 \) from \( 27 \):
\[ 27 - 25 = 2 \]
Now bring down the next pair (66), making it \( 266 \).

### Step 4: Double the number above the line
Double the number above the line (5), which gives \( 10 \). Write this down as a prefix to the next digit we will find.

### Step 5: Find the next digit
We need to find a digit \( x \) such that \( (10x) \times x \) is less than or equal to \( 266 \).
- Trying \( x = 2 \):
\[ (10 \cdot 2) \cdot 2 = 20 \cdot 2 = 40 \quad (too small) \]
- Trying \( x = 3 \):
\[ (10 \cdot 3) \cdot 3 = 30 \cdot 3 = 90 \quad (too small) \]
- Trying \( x = 6 \):
\[ (10 \cdot 6) \cdot 6 = 60 \cdot 6 = 360 \quad (too large) \]
- Trying \( x = 2 \):
\[ (10 \cdot 2) \cdot 2 = 20 \cdot 2 = 40 \quad (too small) \]
- Trying \( x = 4 \):
\[ (10 \cdot 4) \cdot 4 = 40 \cdot 4 = 160 \quad (fits) \]
- Trying \( x = 5 \):
\[ (10 \cdot 5) \cdot 5 = 50 \cdot 5 = 250 \quad (fits) \]
So we take \( x = 4 \).

### Step 6: Subtract and bring down the next pair
Subtract \( 250 \) from \( 266 \):
\[ 266 - 250 = 16 \]
Now bring down the next pair (76), making it \( 1676 \).

### Step 7: Double the number above the line
Now we have \( 54 \) above the line. Double it to get \( 108 \).

### Step 8: Find the next digit
We need to find a digit \( y \) such that \( (108y) \times y \) is less than or equal to \( 1676 \).
- Trying \( y = 1 \):
\[ (108 \cdot 1) \cdot 1 = 108 \quad (fits) \]
- Trying \( y = 2 \):
\[ (108 \cdot 2) \cdot 2 = 216 \quad (fits) \]
- Trying \( y = 3 \):
\[ (108 \cdot 3) \cdot 3 = 324 \quad (fits) \]
- Trying \( y = 4 \):
\[ (108 \cdot 4) \cdot 4 = 432 \quad (fits) \]
- Trying \( y = 5 \):
\[ (108 \cdot 5) \cdot 5 = 540 \quad (fits) \]
- Trying \( y = 6 \):
\[ (108 \cdot 6) \cdot 6 = 648 \quad (fits) \]
- Trying \( y = 7 \):
\[ (108 \cdot 7) \cdot 7 = 756 \quad (fits) \]
- Trying \( y = 8 \):
\[ (108 \cdot 8) \cdot 8 = 864 \quad (fits) \]
- Trying \( y = 9 \):
\[ (108 \cdot 9) \cdot 9 = 972 \quad (fits) \]
- Trying \( y = 10 \):
\[ (108 \cdot 10) \cdot 10 = 1080 \quad (fits) \]
- Trying \( y = 11 \):
\[ (108 \cdot 11) \cdot 11 = 1188 \quad (fits) \]
- Trying \( y = 12 \):
\[ (108 \cdot 12) \cdot 12 = 1296 \quad (fits) \]
- Trying \( y = 13 \):
\[ (108 \cdot 13) \cdot 13 = 1404 \quad (fits) \]
- Trying \( y = 14 \):
\[ (108 \cdot 14) \cdot 14 = 1512 \quad (fits) \]
- Trying \( y = 15 \):
\[ (108 \cdot 15) \cdot 15 = 1620 \quad (fits) \]
- Trying \( y = 16 \):
\[ (108 \cdot 16) \cdot 16 = 1728 \quad (too large) \]
So we take \( y = 15 \).

### Step 9: Final subtraction
Subtract \( 1620 \) from \( 1676 \):
\[ 1676 - 1620 = 56 \]
Now we have \( 56 \) as the remainder.

### Conclusion
The square root of \( 276676 \) is \( 526 \).

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