Show that 6292 is not a perfect square.

Answer

To show that 6292 is not a perfect square, we will use the long division method for square roots. Here’s a step-by-step solution:

### Step 1: Pair the Digits
We start by pairing the digits of 6292 from the right. This gives us the pairs: (62)(92).

### Step 2: Find the Largest Square
Next, we look at the first pair, which is 62. We need to find the largest integer whose square is less than or equal to 62. The largest integer is 7, since \(7^2 = 49\) and \(8^2 = 64\) (which is too large).

### Step 3: Subtract and Bring Down the Next Pair
Now, we subtract \(49\) from \(62\):
\[ 62 - 49 = 13 \]
Next, we bring down the next pair (92), giving us 1392.

### Step 4: Double the Quotient
Now, we double the quotient we have (which is 7) to get 14. We will denote the next digit of the quotient as \(x\) and form the expression \(14x\).

### Step 5: Find the Next Digit
We need to find \(x\) such that \(14x \times x\) is less than or equal to 1392. Testing \(x = 9\):
\[ 14 \times 9 = 126 \]
\[ 126 \times 9 = 1134 \]
This is less than 1392. Now, testing \(x = 10\):
\[ 14 \times 10 = 140 \]
\[ 140 \times 10 = 1400 \]
This is greater than 1392. So, we take \(x = 9\).

### Step 6: Subtract Again
Now we subtract \(1134\) from \(1392\):
\[ 1392 - 1134 = 258 \]
Now we bring down the next pair (00), giving us 25800.

### Step 7: Double the Quotient Again
Now, we double the current quotient (which is 79) to get 158. We will denote the next digit of the quotient as \(y\) and form the expression \(158y\).

### Step 8: Find the Next Digit Again
We need to find \(y\) such that \(158y \times y\) is less than or equal to 25800. Testing \(y = 1\):
\[ 158 \times 1 = 158 \]
\[ 158 \times 1 = 158 \]
This is less than 25800. Testing \(y = 2\):
\[ 158 \times 2 = 316 \]
\[ 316 \times 2 = 632 \]
This is still less than 25800. Testing \(y = 3\):
\[ 158 \times 3 = 474 \]
\[ 474 \times 3 = 1422 \]
This is still less than 25800. Testing \(y = 4\):
\[ 158 \times 4 = 632 \]
\[ 632 \times 4 = 2528 \]
This is still less than 25800. Testing \(y = 5\):
\[ 158 \times 5 = 790 \]
\[ 790 \times 5 = 3950 \]
This is still less than 25800. Testing \(y = 6\):
\[ 158 \times 6 = 948 \]
\[ 948 \times 6 = 5688 \]
This is still less than 25800. Testing \(y = 7\):
\[ 158 \times 7 = 1106 \]
\[ 1106 \times 7 = 7742 \]
This is still less than 25800. Testing \(y = 8\):
\[ 158 \times 8 = 1264 \]
\[ 1264 \times 8 = 10112 \]
This is still less than 25800. Testing \(y = 9\):
\[ 158 \times 9 = 1422 \]
\[ 1422 \times 9 = 12798 \]
This is still less than 25800. Testing \(y = 10\):
\[ 158 \times 10 = 1580 \]
\[ 1580 \times 10 = 15800 \]
This is still less than 25800. Testing \(y = 11\):
\[ 158 \times 11 = 1738 \]
\[ 1738 \times 11 = 19118 \]
This is still less than 25800. Testing \(y = 12\):
\[ 158 \times 12 = 1896 \]
\[ 1896 \times 12 = 22752 \]
This is still less than 25800. Testing \(y = 13\):
\[ 158 \times 13 = 2054 \]
\[ 2054 \times 13 = 26662 \]
This is greater than 25800. So, the largest \(y\) we can use is 12.

### Conclusion
Since we cannot find a perfect square that equals 6292, we conclude that 6292 is not a perfect square.

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