By what least number should the given number be divided to get a perfect square number?

(i) Resolving 1575 into prime factors:

$1575=3 \times 3 \times 5 \times 5 \times 7=3^{2} \times 5^{2} \times 7$

Thus, to get a perfect square, the given number should be divided by 7

New number obtained $=\left(3^{2} \times 5^{2}\right)=(3 \times 5)^{2}=(15)^{2}$

Hence, the new number is the square of 15

(ii) Resolving 9075 into prime factors: 

$9075=3 \times 5 \times 5 \times 11 \times 11=3 \times 5^{2} \times 11^{2}$

Thus, to get a perfect square, the given number should be divided by 3

New number obtained $=\left(5^{2} \times 11^{2}\right)=(5 \times 11)^{2}=(55)^{2}$

Hence, the new number is the square of  55

(iii) Resolving 4851 into prime factors:

$4851=3 \times 3 \times 7 \times 7 \times 11=3^{2} \times 7^{2} \times 11$

Thus, to get a perfect square, the given number should be divided by 11

New number obtained $=\left(3^{2} \times 7^{2}\right)=(3 \times 7)^{2}=(21)^{2}$

Hence, the new number is the square of 21

(iv) Resolving 3380 into prime factors: 

$3380=2 \times 2 \times 5 \times 13 \times 13=2^{2} \times 5 \times 13^{2}$

Thus, to get a perfect square, the given number should be divided by 5

New number obtained $=\left(2^{2} \times 13^{2}\right)=(2 \times 13)^{2}=(26)^{2}$

Hence, the new number is the square of 26

(v) Resolving 4500 into prime factors: 

$4500=2 \times 2 \times 3 \times 3 \times 5 \times 5 \times 5=2^{2} \times 3^{2} \times 5^{2} \times 5$

Thus, to get a perfect square, the given number should be divided by 5

New number obtained $=\left(2^{2} \times 3^{2} \times 5^{2}\right)=(2 \times 3 \times 5)^{2}=(30)^{2}$

Hence, the new number is the square of 30

(vi) Resolving 7776 into prime factors:

$7776=2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3=2^{2} \times 2^{2} \times 2 \times 3^{2} \times 3^{2} \times 3$

Thus, to get a perfect square, the given number should be divided by 6 whish is a product of 2 and 3

New number obtained $=\left(2^{2} \times 2^{2} \times 3^{2} \times 3^{2}\right)=(2 \times 2 \times 3 \times 3)^{2}=(36)^{2}$

Hence, the new number is the square of 36

(vii) Resolving 8820 into prime factors:

$8820=2 \times 2 \times 3 \times 3 \times 5 \times 7 \times 7=2^{2} \times 3^{2} \times 5 \times 7^{2}$

New number obtained $=\left(2^{2} \times 3^{2} \times 7^{2}\right)=(2 \times 3 \times 7)^{2}=(42)^{2}$

Hence, the new number is the square of 42

(viii) Resolving 4056 into prime factors: 

$4056=2 \times 2 \times 2 \times 3 \times 13 \times 13=2^{2} \times 2 \times 3 \times 13^{2}$

Thus, to get a perfect square, the given number should be divided by 6, which is a product of 2 and 3

New number obtained $=\left(2^{2} \times 13^{2}\right)=(2 \times 13)^{2}=(26)^{2}$

Hence, the new number is the square of 26