### Question 1. Find the square root of each of the following by long division method:

### (i) 12544

**Solution:**

Here, by using long division method,

We found that the square root of 12544 is 112.

### (ii) 97344

**Solution:**

Here, by using long division method

We found that the square root of 97344 is 312.

### (iii) 286225

**Solution:**

Here, by using long division method,

We found that the square root of 286225 is 535.

### (iv) 390625

**Solution:**

Here, by using long division method,

We found that the square root of 390625 is 625.

### (v) 363609

**Solution:**

Here, by using long division method,

We found that the square root of 363609 is 603.

### (vi) 974169

**Solution:**

Here, by using long division method,

We found that the square root of 974169 is 987.

### (vii) 120409

**Solution:**

Here, by using long division method,

We found that the square root of 120409 is 347.

### (viii) 1471369

**Solution:**

Here, by using long division method,

We found that the square root of 1471369 is 1213.

### (ix) 291600

**Solution: **

Here, by using long division method,

We found that the square root of 291600 is 540.

### (x) 9653449

**Solution:**

Here, by using long division method,

We found that the square root of 9653449 is 3107.

### (xi) 1745041

**Solution:**

Here, by using long division method,

We found that the square root of 1745041 is 1321.

### (xii) 4008004

**Solution:**

Here, by using long division method,

We found that the square root of 4008004 is 2002.

### (xiii) 20657025

**Solution: **

Here, by using long division method,

We found that the square root of 20657025 is 4545.

### (xiv) 152547201

**Solution:**

Here, by using long division method,

We found that the square root of 152547201 is 12351.

### (xv) 20421361

**Solution: **

Here, by using long division method,

We found that the square root of 20421361 is 4519.

### (xvi) 62504836

**Solution:- **

Here, by using long division method,

We found that the square root of 62504836 is 7906.

### (xvii) 82264900

**Solution:**

Here, using long division method,

We found that the square root of 82264900 is 9070.

### (xviii) 3226694416

**Solution:**

Here, by using long division method,

We found that the square root of 3226694416 is 56804.

### (xix) 6407522209

**Solution:**

Here, by using long division method,

We found that the square root of 6407522209 is 80047.

### (xx) 3915380329

**Solution: **

Here, by using long division method,

We found that the square root of 3915380329 is 62573.

### Question 2. Find the least number which must be subtracted from the following numbers to make them a perfect square:

### (i) 2361

**Solution:**

Using long division method,

Here, we observed that 2361 is 57 more than the square of 48.

So, we have to subtract 57 from 2361 to make it a perfect square.

### (ii) 194491

**Solution:**

Using long division method,

Here, we observed that 194491 is 10 more than the square of 441.

So, we have to subtract 10 from 194491 to make it a perfect square.

### (iii) 26535

**Solution:**

using long division method,

Here, we observed that 26535 is 291 more than the square of 162.

So, we have to subtract 291 from 26535 to make it a perfect square.

**(iv) 161605**

**Solution:**

Using long division method,

Here, we observed that 161605 is 1 more than the square of 402.

So, we have to subtract 1 from 161605 to make it a perfect square.

### (v) 4401624

**Solution:**

Using long division method,

Here, we observed that 4401624 is 20 more than the square of 2098.

So, we have to subtract 20 from 4401624 to make it a perfect square.

### Question 3. Find the least number which must be added to the following numbers to make them a perfect square:

### (i) 5607

**Solution:**

Here, by using long division method,

Now, we observe that 5607 is 18 less than square of 75.

Therefore, we must add 18 to 5607 (18 + 5607 = 5625) to get a perfect square of 75 which is 5625.

### (ii) 4931

**Solution:**

Here, by using long division method,

Now, we observe that 4931 is 110 less than square of 71.

Therefore, we must add 110 to 4931 (110 + 4931 = 5041) to get a perfect square of 71 which is 5041.

### (iii) 4515600

**Solution:**

Here, by using long division method,

Now, we observe that 4515600 is 25 less than square of 2125.

Therefore, we must add 18 to 4515600 (25 + 4515600 = 4515625) to get a perfect square of 2125 which is 4515625.

### (iv) 37460

**Solution:**

Here, by using long division method,

Now, we observe that 37460 is 176 less than square of 194.

Therefore, we must add 176 to 37460 (176 + 37460 = 37636) to get a perfect square of 194 which is 37636.

### (v) 506900

**Solution:**

Here, by using long division method,

Now, we observe that 506900 is 44 less than square of 712.

Therefore, we must add 44 to 506900 (44 + 506900 = 506944) to get a perfect square of 712 which is 506944.

### Question 4. Find the greatest number of 5 digits which is a perfect square.

**Solution:**

We know that 99999 is the greatest 5-digit number.

Now, to get the greatest 5 digits square number, we have to find the smallest

number that can be subtracted to obtain a perfect square.

So, By using long division method,

Now, the remainder here is 143.

So, 143 must be subtracted from 99999 to get a perfect square.

So, 99999 – 143 = 99856.

Therefore, 99856 is the greatest 5 digits which is a perfect square.

### Question 5. Find the least number of 4 digits which is a perfect square.

**Solution: **

We know that 1000 is the smallest 4-digit number.

Now, in order to obtain the least 4 digits perfect square number, we have to find the

smallest number that can be added to obtain a perfect square.

So, By using long division method,

Now, we have to add 24 to 1000 to get a perfect square number.

So, 24 + 1000 = 1024

Therefore, 1024 is the least perfect square number with 4 digits which is a square of 32.

### Question 6. Find the least number of six digits which is a perfect square.

**Solution:**

We know that 100000 is the smallest 6 digit number.

Now, in order to obtain the least 6 digits perfect square number, we have to find

the smallest number that can be added to obtain a perfect square.

So, By using long division method,

Now, we have to add 489 to 100000 to get a perfect square number.

So, 489 + 100000 = 100489

Therefore, 100489 is the least perfect square number with 6 digits which is a square of 317.

### Question 7. Find the greatest number of 4 digits which is a perfect square.

**Solution:**

We know that 9999 is the greatest 4 digit number.

Now, to get the greatest 4 digits square number, we have to find the smallest number

that can be subtracted to obtain a perfect square.

So, By using long division method,

Now, the remainder here is 198.

So, 198 must be subtracted from 9999 to get a perfect square.

So, 9999 – 198 = 9801.

Therefore, 9801 is the greatest 4 digits which is a perfect square number.

### Question 8. A General arranges his soldiers in rows to form a perfect square. He finds that in doing so, 60 soldiers are left out. If the total number of soldiers be 8160, find the number of soldiers in each row?

**Solution:**

Given, Total number of Soldiers = 8160

Number of left out Soldiers = 60

So, number of soldiers arranged in rows to form a perfect square is 8100 (8160 – 60 = 8100)

So, the number of soldiers in each row = square root of 8100

So, √8100 = 90

Therefore, there are 90 soldiers in each row.

### Question 9. The area of a square field is 60025 m^{2}. A man cycles along its boundary at 18 Km/hr. In how much time will he return at the starting point?

**Solution:**

Given, area of square field = 60025 m

^{2}Speed of the cyclist = 18 km/h

= 18 × (1000/60 × 60)

= 5 m/s

^{2 }Area of square field = 60025 m

^{2}or, (side)

^{2 }= 60025or, side = √60025

or, side = 245 m

We know that, Total length of boundary = perimeter

= 4 * side (Perimeter of square = 4 * side of square)

= 4 × 245

= 980 m

Hence, Time taken to return to the starting point = 980/5 (time = distance / speed)

= 196 seconds

= 3 minutes 16 seconds

### Question 10. The cost of leveling and turning a square lawn at Rs 2.50 per m^{2} is Rs13322.50. Find the cost of fencing it at Rs 5 per metre.

**Solution:**

Given, The cost of leveling and turning a square lawn = Rs 2.50 per m

^{2}Total cost of leveling and turning a square lawn = Rs13322.50

Now, Area of square lawn = total cost of leveling and turning a square lawn / the cost of leveling and turning a square lawn

= 13322.50 / 2.50

= 5329 m

^{2}So, the length of side of square = √area of square (As area of square = side

^{2})= √5329 = 73 m

So, Total length for fencing = Perimeter (As perimeter of square = 4 * side)

= 4 * 73 = 292 m

Therefore, cost of fencing the square lawn = 5 * 292

= Rs. 1460

### Question 11. Find the greatest number of three digits which is a perfect square.

**Solution:**

We know that 999 is the greatest 3-digit number.

Now, to get the greatest 3 digits square number, we have to find the smallest number

that can be subtracted to obtain a perfect square.

So, By using long division method,

Now, the remainder here is 38.

So, 38 must be subtracted from 999 to get a perfect square.

So, 999 – 38 = 961.

961 = 31

^{2}Therefore, 961 is the greatest 3 digits which is a perfect square number.

### Question 12. Find the smallest number which must be added to 2300 so that it becomes a perfect square.

**Solution:**

We will first find the square root of the given number by using the long division method,

Now, we come to know that 2300 is 4 less than square of 48.

So, 48

^{2}= 2304Hence, 4 must be added to 2300 in order to get a perfect square.