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

Question:

By what least number should the given number be divided to get a perfect square number? In each case, find the number whose square is the new number.

(i) 1575

(ii) 9075

(iii) 4851

(iv) 3380

(v) 4500

(vi) 7776

(vii) 8820

(viii) 4056

Solution:

(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

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