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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

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Solution

1. Resolving 3675 into prime factors:
3675=3×5×5×7×7

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

New number= (32×52×72)=(3×5×7)2=(105)2

Hence, the new number is the square of 105.

2. Resolving 2156 into prime factors:
2156=2×2×7×7×11=(22×72×11)

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

New number =(22×72×112)=(2×7×11)2=(154)2

Hence, the new number is the square of 154.

3. Resolving 3332 into prime factors:
3332=2×2×7×7×17=22×72×17

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

New number =(22×72×172)=(2×7×17)2=(238)2

Hence, the new number is the square of 238.

4. Resolving 2925 into prime factors:
2925=3×3×5×5×13=32×52×13

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

New number =(32×52×132)=(3×5×13)2=(195)2

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

5. Resolving 9075 into prime factors:

9075=3×5×5×11×11=3×52×112

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

New number =(32×52×112)=(3×5×11)2=(165)2

Hence, the new number is the square of 165.

6. Resolving 7623 into prime factors:
7623=3×3×7×11×11=32×7×112

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

New number =(32×72×112)=(3×7×11)2=(231)2

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

7. Resolving 3380 into prime factors:
3380=2×2×5×13×13=22×5×132

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

New number =(22×52×132)=(2×5×13)2=(130)2

Hence, the new number is the square of 130.

8. Resolving 2475 into prime factors:
2475=3×3×5×5×11=32×52×11

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

New number =(32×52×112)=(3×5×11)2=(165)2

Hence, the new number is the square of 165.

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