b) Let's take a number 260 . (i) Find the square root of 260 by division method. (ii) What is the remainder left in this process? (iii) Subtract the remainder from 260 . Is the difference a perfect square? Section - B

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Let's address each part of the question step by step.

### **Given Number:** 260

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### **(i) Find the Square Root of 260 by Division Method**

**Step-by-Step Process:**

1. **Pair the Digits:**  
   Start from the decimal point and pair the digits of 260.  
   $$ \text{Pairs: } 2 \ | \ 60 $$

2. **Find the Largest Square ≤ First Pair (2):**  
   $$ 1^2 = 1 \leq 2 $$  
   So, the first digit of the square root is **1**.  
   Subtract:  
   $$ 2 - 1 = 1 $$

3. **Bring Down the Next Pair (60):**  
   Combine with the remainder: **160**.

4. **Double the Current Root (1 × 2 = 2) and Find the Next Digit (d):**  
   We need $$ 20 + d $$ such that $$ (20 + d) \times d \leq 160 $$.

- $$ d = 6 $$: $$ 26 \times 6 = 156 \leq 160 $$
   - $$ d = 7 $$: $$ 27 \times 7 = 189 > 160 $$ (Too big)

So, **d = 6**.  
   Add to the root: **16**.  
   Subtract:  
   $$ 160 - 156 = 4 $$

5. **Optional (For More Precision):**  
   If desired, you can continue by bringing down pairs of zeros after the decimal point to find more decimal places. However, for the purpose of this problem, we'll stop here.

**Final Approximation:**  
$$ \sqrt{260} \approx 16 \text{ with a remainder of } 4 $$

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### **(ii) What is the Remainder Left in This Process?**

After finding that $$ 16^2 = 256 $$ and subtracting from 260:

$$ 260 - 256 = 4 $$

**Remainder:** **4**

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### **(iii) Subtract the Remainder from 260. Is the Difference a Perfect Square?**

**Calculation:**  
$$ 260 - 4 = 256 $$

**Verification:**  
$$ 16^2 = 256 $$

**Conclusion:**  
Yes, **256 is a perfect square**.

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### **Summary of Answers:**

1. **Square Root of 260:**  
   $$ \sqrt{260} \approx 16 \text{ with a remainder of } 4 $$

2. **Remainder Left:**  
   $$ 4 $$

3. **Difference $$260 - 4 = 256$$:**  
   $$ 256 \text{ is a perfect square (}16^2\text{)} $$

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- **Square Root of 260:** Approximately 16 with a remainder of 4. - **Remainder:** 4 - **Difference (260 - 4):** 256, which is a perfect square (16²).

b) Let’s take a number 260 .
(i) Find the square root of 260 by division method.
(ii) What is the remainder left in this process?
(iii) Subtract the remainder from 260 . Is the difference a perfect square?
Section - B

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b) Let’s take a number 260 . (i) Find the square root of 260 by division method. (ii) What is the remainder left in this process? (iii) Subtract the remainder from 260 . Is the difference a perfect square? Section - B