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

Question:

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

(i) 3975

(ii) 2156

(iii) 3332

(iv) 2925

(v) 9075

(vi) 7623

(vii) 3380

(viii) 2475

Solution:

1. Resolving 3675 into prime factors:

$3675=3 \times 5 \times 5 \times 7 \times 7$

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

New number $=\left(3^{2} \times 5^{2} \times 7^{2}\right)=(3 \times 5 \times 7)^{2}=(105)^{2}$

Hence, the new number is the square of 105.

2. Resolving 2156 into prime factors:

$2156=2 \times 2 \times 7 \times 7 \times 11=\left(2^{2} \times 7^{2} \times 11\right)$

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

New number $=\left(2^{2} \times 7^{2} \times 11^{2}\right)=(2 \times 7 \times 11)^{2}=(154)^{2}$

Hence, the new number is the square of 154.

3. Resolving 3332 into prime factors:

$3332=2 \times 2 \times 7 \times 7 \times 17=2^{2} \times 7^{2} \times 17$

Thus, to get a perfect square, the given number should be multiplied by 17.

New number $=\left(2^{2} \times 7^{2} \times 17^{2}\right)=(2 \times 7 \times 17)^{2}=(238)^{2}$

Hence, the new number is the square of 238.

4. Resolving 2925 into prime factors:

$2925=3 \times 3 \times 5 \times 5 \times 13=3^{2} \times 5^{2} \times 13$

Thus, to get a perfect square, the given number should be multiplied by 13.

New number $=\left(3^{2} \times 5^{2} \times 13^{2}\right)=(3 \times 5 \times 13)^{2}=(195)^{2}$

Hence, the number whose square is the new number is 195.

5. 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 multiplied by 3.

New number $=\left(3^{2} \times 5^{2} \times 11^{2}\right)=(3 \times 5 \times 11)^{2}=(165)^{2}$

Hence, the new number is the square of 165.

6. Resolving 7623 into prime factors:

$7623=3 \times 3 \times 7 \times 11 \times 11=3^{2} \times 7 \times 11^{2}$

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

New number $=\left(3^{2} \times 7^{2} \times 11^{2}\right)=(3 \times 7 \times 11)^{2}=(231)^{2}$

Hence, the number whose square is the new number is 231.

7. 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 multiplied by 5.

New number $=\left(2^{2} \times 5^{2} \times 13^{2}\right)=(2 \times 5 \times 13)^{2}=(130)^{2}$

Hence, the new number is the square of 130.

8. Resolving 2475 into prime factors:

$2475=3 \times 3 \times 5 \times 5 \times 11=3^{2} \times 5^{2} \times 11$

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

New number $=\left(3^{2} \times 5^{2} \times 11^{2}\right)=(3 \times 5 \times 11)^{2}=(165)^{2}$

Hence, the new number is the square of 165.

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