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?

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Let's solve each part of the problem step by step.

### **b) Given Number: 260**

#### **(i) Find the square root of 260 by the division method.**

The **division method** for finding square roots involves a process similar to long division. Here's how to apply it to find √260:

1. **Pair the Digits:**
   - Start from the decimal point and pair the digits in twos: **2 | 60**

2. **Find the Largest Square ≤ First Pair (2):**
   - The largest integer whose square is ≤ 2 is **1** (since $$1^2 = 1$$).
   - **Quotient so far:** 1
   - **Remainder:** $$2 - 1 = 1$$

3. **Bring Down the Next Pair (60):**
   - Combine the remainder with the next pair: $$1$$ (remainder) becomes **160**.

4. **Double the Current Quotient and Find the Next Digit:**
   - Double the current quotient (1): $$2 \times 1 = 2$$.
   - Find a digit **x** such that $$(20 + x) \times x ≤ 160$$.
     - Testing **x = 6**: $$26 \times 6 = 156 ≤ 160$$
     - **Next digit:** 6
   - **Quotient so far:** 16
   - **Remainder:** $$160 - 156 = 4$$

5. **(Optional) Continue for Decimal Precision:**
   - If you need more decimal places, continue the process by bringing down pairs of zeros and repeating the steps. For simplicity, we'll stop here.

**Approximate Square Root:** $$ \sqrt{260} \approx 16 $$ (to the nearest integer)

#### **(ii) What is the remainder left in this process?**

- **Remainder:** 4

#### **(iii) Subtract the remainder from 260. Is the difference a perfect square?**

- **Calculate the Difference:**
  $$
  260 - 4 = 256
  $$
  
- **Check if 256 is a Perfect Square:**
  $$
  16^2 = 256
  $$
  
  Yes, **256** is a perfect square since $$ 16^2 = 256 $$.

### **Summary:**
- **(i)** The square root of 260 is **16** with a remainder.
- **(ii)** The **remainder** is **4**.
- **(iii)** Subtracting the remainder from 260 gives **256**, which is a **perfect square** ($$16^2$$).
- **(i)** The square root of 260 is approximately 16 with a remainder of 4. - **(ii)** The remainder is 4. - **(iii)** Subtracting the remainder from 260 gives 256, which is a perfect square.

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?

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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?