What is the least number to be added to $7700$ to make it a perfect square?

### Concept & Strategy When finding what to *add* to a number to reach a perfect square, use the long division square root method to find the integer root just below the target number. Then, calculate the square of the *next* integer and find the difference. ### Step-by-Step Solution Let's find the integer square root of $7700$ using long division. **Step 1:** Group the digits: $\overline{77} \ \overline{00}$. **Step 2:** Find the nearest square to $77$. $8^2 = 64$. $77 - 64 = 13$. Bring down the $00$, making the new dividend $1300$. Prefix is $8$. **Step 3:** Double the prefix $8$ to $16$. We need $16x \times x \le 1300$. $167 \times 7 = 1169$. $168 \times 8 = 1344$. (This is larger than 1300, so $x$ must be $7$ for subtraction, but we want the *next* square). Since $167 \times 7 = 1169$, the integer square root is $87$ with a remainder. This means: $87^2 < 7700$ To find what to *add*, we must find the value of the *next* perfect square, which is $88^2$. $88^2 = (80 + 8)^2 = 6400 + 1280 + 64 = 7744$. Calculate the required addition: Amount to add $= 7744 - 7700 = 44$. Comparing $44$ with the given options (77, 98, 131, 221), it is not listed. ### Exam Strategy & Shortcut Instead of full long division, you can estimate. You know $80^2 = 6400$ and $90^2 = 8100$. The number $7700$ is closer to $90^2$. Check $85^2 = 7225$. Too small. Check $88^2 = 7744$. Check $87^2 = 7569$. Since $7700$ is between $87^2$ and $88^2$, the next perfect square is exactly $7744$. Subtracting $7700$ gives $44$. Takes $15$ seconds if you know your squares. ### Common Pitfall Finding the remainder for what to *subtract* ($1300 - 1169 = 131$) and selecting option (c). The question explicitly asks what must be *added*, meaning you must aim for the perfect square immediately *above* the given number. ### Final Answer Therefore, the correct answer is **None of these**.