**Question 1. Which of the following numbers are perfect squares?**

**(i) 484**

**Solution:**

Prime factorization of 484 is

484 = 2×2×11×11

By grouping prime factors

= (2×2) × (11×11)

Proper pairing of all the factors

Therefore, 484 is a perfect square.

**(ii) 625**

**Solution:**

Prime factorization of 625

625 = 5×5×5×5

By grouping the prime factors

= (5×5) × (5×5)

Proper pairing of all the factors

Therefore, 625 is a perfect square.

**(iii) 576**

**Solution:**

Prime factorization of 576

576 = 2×2×2×2×2×2×3×3

By grouping the prime factors

= (2×2) × (2×2) × (2×2) × (3×3)

Proper pairing of all the factors

Therefore, 576 is a perfect square.

**(iv) 941**

**Solution:**

Prime factorization of 941

941 = 941 × 1

941 itself is a prime factor.

Therefore, 941 is not a perfect square.

**(v) 961**

**Solution:**

Prime factorization of 961

961 = 31×31

By grouping the prime factors

= (31×31)

Proper pairing of all factors

Therefore, 961 is a perfect square.

**(vi) 2500**

**Solution:**

Prime factorization of

2500 = 2×2×5×5×5×5

By grouping the prime factors

= (2×2) × (5×5) × (5×5)

Proper pairing of all factors

Therefore, 2500 is a perfect square.

**Question 2. Show that each of the following numbers is a perfect square. Also**,** find the number whose square is the given number in each case:**

**(i) 1156**

**Solution:**

Prime factorization of 1156

1156 = 2×2×17×17

By grouping prime factors

= (2×2) × (17×17)

Proper pairing of all factors

Therefore,1156 is a perfect square.

To find whose square the given number is

1156 = (2×17) × (17×2)

= 34 × 34

= (34)

^{2}Therefore,1156 is a square of 34.

**(ii) 2025**

**Solution:**

Prime factorization of 2025

2025 = 3×3×3×3×5×5

By grouping prime factors

= (3×3) × (3×3) × (5×5)

Proper pairing of all factors

Therefore, 2025 is a perfect square.

To find whose square the given number is

2025 = (3×3×5) × (3×3×5)

= 45 × 45

= (45)

^{2}Therefore, 2025 is a square of 45.

**(iii) 14641**

**Solution:**

Prime factorization of 14641

14641 = 11×11×11×11

By grouping prime factors

= (11×11) × (11×11)

Proper pairing of all factors

Therefore,14641 is a perfect square.

To find whose square the given number is

14641 = (11×11) × (11×11)

= 121 × 121

= (121)

^{2}Therefore,14641 is a square of 121.

**(iv) 4761**

**Solution:**

Prime factorization of 4761

4761 = 3×3×23×23

By grouping the prime factors

= (3×3) × (23×23)

Proper pairing of all factors

Therefore, 4761 is a perfect square.

To find whose square the given number is

4761 = (3×23) × (3×23)

= 69 × 69

= (69)

^{2}Therefore, 4761 is a square of 69.

**Question 3. Find the smallest number by which the given number must be multiplied so that the product is a perfect square:**

**(i) 23805**

**Solution:**

Prime factorization of 23805

23805 = 3×3×23×23×5

By grouping the prime factors

= (3×3) × (23×23) × 5

Prime factor 5 is left out.

Multiplying by 5,

23805 × 5 = (3×3) × (23×23) × (5×5)

= (3×5×23) × (3×5×23)

= 345 × 345

= (345)

^{2}Therefore, product is the square of 345.

**(ii) 12150**

**Solution:**

Prime factorization of 12150

12150 = 2×3×3×3×3×3×5×5

By grouping the prime factors

= 2×3 × (3×3) × (3×3) × (5×5)

Prime factor 2 and 3 are left out.

Multiplying by 2×3 = 6

12150 × 6 = 2×3 × (3×3) × (3×3) × (5×5) × 2 × 3

= (2×3×3×3×5) × (2×3×3×3×5)

= 270 × 270

= (270)

^{2}Therefore, product is the square of 270.

**(iii) 7688**

**Solution:**

Prime factorization of 7688

7688 = 2×2×31×31×2

By grouping the prime factors

= (2×2) × (31×31) × 2

Prime factor 2 is left out.

Multiplying by 2

7688 × 2 = (2×2) × (31×31)× (2×2)

= (2×31×2) × (2×31×2)

= 124 × 124

= (124)

^{2}Therefore, product is the square of 124.

**Question 4. Find the smallest number by which the given number must be divided so that the resulting number is a perfect square:**

**(i) 14283**

**Solution:**

Prime factorization of 14283

14283 = 3×3×3×23×23

By grouping the prime factors

= (3×3) × (23×23) × 3

Prime factor 3 is left out.

Dividing by 3

14283/3 = (3×3) × (23×23)

= (3×23) × (3×23)

= 69 × 69

= (69)

^{2}Therefore, resultant is the square of 69.

**(ii) 1800**

**Solution:**

Prime factorization of 1800

1800 = 2×2×5×5×3×3×2

By grouping the prime factors

= (2×2) × (5×5) × (3×3) × 2

Prime factor 2 is left out.

Dividing by 2

1800/2 = (2×2) × (5×5) × (3×3)

= (2×5×3) × (2×5×3)

= 30 × 30

= (30)

^{2}Therefore, resultant is the square of 30.

**(iii) 2904**

**Solution:**

Prime factorization of 2904

2904 = 2×2×11×11×2×3

By grouping the prime factors

= (2×2) × (11×11) × 2 × 3

Prime factor 2 and 3 are left out.

Dividing by 6

2904/6 = (2×2) × (11×11)

= (2×11) × (2×11)

= 22 × 22

= (22)

^{2}Therefore, resultant is the square of 22.

**Question 5. Which of the following numbers are perfect squares?**

**11, 12, 16, 32, 36, 50, 64, 79, 81, 111, 121**

**Solution:**

11 = 1×11

11 is a prime number

Therefore, 11 is not a perfect square.

12 = 2×2×3

3 is left out in pairing of factors

Therefore, 12 is not a perfect square.

16 = 2×2×2×2

= (2×2) × (2×2)

Proper pairing of factors

Therefore,16 is a perfect square.

32 = 2×2×2×2×2

= (2×2) × (2×2) × 2

2 is left out in pairing of factors

Therefore, 32 is not a perfect square.

36 = 2×2×3×3

Proper pairing of factors

Therefore, 36 is a perfect square.

50 = 2×5×5

2 is left out in pairing of factors

Therefore, 50 is not a perfect square.

64 = 2×2×2×2×2×2

Proper pairing of factors

Therefore, 64 is a perfect square.

79 = 1×79

79 is a prime number

79 is not a perfect square.

81 = 3×3×3×3

Proper pairing of factors

Therefore, 81 is a perfect square.

111 = 1×111

111 it is a prime number.

Therefore, 111 is not a perfect square.

121 = 11×11

Proper pairing of factors

Therefore, 121 is a perfect square.

**Question 6. Using prime factorization method, find which of the following numbers are perfect squares?**

**189, 225, 2048, 343, 441, 2961, 11025, 3549**

**Solution:**

189 prime factors are

189 = 3×3×3×7

As, proper pairing of factors is not there.

Therefore, it is not a perfect square

225 prime factors are

225 = (5×5) × (3×3)

Proper pairing of factors is there.

Therefore, it is a perfect square.

2048 prime factors are

2048 = (2×2) × (2×2) × (2×2) × (2×2) × (2×2) × 2

2 is left out from pairing of factors.

Therefore, 2048 is not a perfect square.

343 prime factors are

343 = (7×7) × 7

7 is left out from proper pairing of factors.

Therefore, 343 is not a perfect square.

441 prime factors are

441 = (7×7) × (3×3)

Proper pairing of factors is there.

Therefore, 441 is a perfect square.

2961 prime factors are

2961 = (3×3) × (3×3) × (3×3) × (2×2)

Proper pairing of factors is there.

Therefore, 2961 is a perfect square.

11025 prime factors are

11025 = (3×3) × (5×5) × (7×7)

Proper pairing of factors is there.

Therefore, 11025 is a perfect square.

3549 prime factors are

3549 = (13×13) × 3 × 7

Proper pairing of factors is not there.

Therefore, 3549 is not a perfect square.

**Question 7. By what number should each of the following numbers by multiplied to get a perfect square in each case? Also find the number whose square is the new number.**

**(i) 8820**

**Solution:**

Prime factorization of 8820

8820 = 2×2×3×3×7×7×5

By grouping the prime factors

= (2×2) × (3×3) × (7×7) × 5

Prime factor 5 is left out.

Multiplying by 5

8820 × 5 = (2×2) × (3×3) × (7×7) × (5×5)

= (2×3×7×5) × (2×3×7×5)

= 210 × 210

= (210)

^{2}Therefore, new number is square of 210.

**(ii) 3675**

**Solution:**

Prime factorization of 3675

3675 = 5×5×7×7×3

By grouping the prime factors

= (5×5) × (7×7) × 3

Prime factor 3 is left out.

Multiplying by 3

3675 × 3 = (5×5) × (7×7) × (3×3)

= (5×7×3) × (5×7×3)

= 105 × 105

= (105)

^{2}Therefore, new number is the square of 105.

**(iii) 605**

**Solution:**

Prime factorization of 605

605 = 5×11×11

By grouping the prime factors

= (11×11) × 5

Prime factor 5 is left out.

Multiply by 5

605 × 5 = (11×11) × (5×5)

= (11×5) × (11×5)

= 55 × 55

= (55)

^{2}Therefore, new number is the square of 55.

**(iv) 2880**

**Solution:**

Prime factorization of 2880

2880 = 5×3×3×2×2×2×2×2×2

By grouping the prime factors

= (3×3) × (2×2) × (2×2) × (2×2) × 5

Prime factor 5 is left out.

Multiply by 5

2880 × 5 = (3×3) × (2×2) × (2×2) × (2×2) × (5×5)

= (3×2×2×2×5) × (3×2×2×2×5)

= 120 × 120

= (120)

^{2}Therefore, new number is the square of 120.

**(v) 4056**

**Solution:**

Prime factorization of 4056

4056 = 2×2×13×13×2×3

By grouping the prime factors

= (2×2) × (13×13) × 2 × 3

Prime factors 2 and 3 are left out.

Multiplying by 6 we get,

4056 × 6 = (2×2) × (13×13) × (2×2) × (3×3)

= (2×13×2×3) × (2×13×2×3)

= 156 × 156

= (156)

^{2}Therefore, new number is the square of 156.

**(vi) 3468**

**Solution:**

Prime factorization of 3468

3468 = 2×2×17×17×3

By grouping the prime factors

= (2×2) × (17×17) × 3

Prime factor 3 is left out.

Multiplying by 3 we get,

3468 × 3 = (2×2) × (17×17) × (3×3)

= (2×17×3) × (2×17×3)

= 102 × 102

= (102)

^{2}Therefore, new number is the square of 102.

**(vii) 7776**

**Solution:**

Prime factorization of 7776

7776 = 2×2×2×2×3×3×3×3×2×3

By grouping the prime factors

= (2×2) × (2×2) × (3×3) × (3×3) × 2 × 3

Prime factors 2 and 3 are left out.

Multiplying by 6 we get,

7776 × 6 = (2×2) × (2×2) × (3×3) × (3×3) × (2×2) × (3×3)

= (2×2×3×3×2×3) × (2×2×3×3×2×3)

= 216 × 216

= (216)

^{2}Therefore, the new number is square of 216.

**Question 8. By What numbers should each of the following be divided to get a perfect square in each case? Also, find the number whose square is the new number.**

**(i) 16562**

**Solution:**

Prime factorization of16562

16562 = 7×7×13×13×2

By grouping the prime factors

= (7×7) × (13×13) × 2

Prime factor 2 is left out.

Dividing by 2

16562/2 = (7×7) × (13×13)

= (7×13) × (7×13)

= 91 × 91

= (91)

^{2}Therefore, the new number is square of 91.

**(ii) 3698**

**Solution:**

Prime factorization of 3698

3698 = 2×43×43

By grouping the prime factors

= (43×43) × 2

Prime factor 2 is left out.

Dividing by 2

3698/2 = (43×43)

= (43)

^{2}Therefore, the new number is square of 43.

**(iii) 5103**

**Solution:**

Prime factorization of 5103

5103 = 3×3×3×3×3×3×7

By grouping the prime factors

= (3×3) × (3×3) × (3×3) × 7

Prime factor 7 is left out.

Dividing by 7

5103/7 = (3×3) × (3×3) × (3×3)

= (3×3×3) × (3×3×3)

= 27 × 27

= (27)

^{2}Therefore, new number is the square of 27.

**(iv) 3174**

**Solution:**

Prime factorization of 3174

3174 = 2×3×23×23

By grouping the prime factors

= (23×23) × 2 × 3

Prime factor 2 and 3 are left out.

Dividing by 6

3174/6 = (23×23)

= (23)

^{2}Therefore, new number is the square of 23.

**(v) 1575**

**Solution:**

Prime factorization for 1575

1575 = 3×3×5×5×7

By grouping the prime factors

= (3×3) × (5×5) × 7

Prime factor 7 is left out.

Dividing by 7

1575/7 = (3×3) × (5×5)

= (3×5) × (3×5)

= 15 × 15

= (15)

^{2}Therefore, new number is square of 15.