# For each of the following numbers, find the smallest whole number by which it should be multiplied so as to get a perfect square number.

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For each of the following numbers, find the smallest whole number by which it should be multiplied so as to get a perfect square number. Also find the square root of the square number so obtained.

(i) 1458  (ii) 768

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(i) 1458 can be factorised as follows.

 2 1458 3 729 3 243 3 81 3 27 3 9 3 3 1

1458 = 2 × 3 × 3 × 3 × 3 × 3 × 3

Here, prime factor 2 does not have its pair. If 2 gets a pair, then the number will become a perfect square. Therefore, 1458 has to be multiplied with 2 to obtain a perfect square.

Therefore, 1458 × 2 = 2916 is a perfect square.

1458 × 2 = 2916 = 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3

∴ √2916 = 2 x 3 x 3 x 3 = 54

(ii) 768 can be factorised as follows.

 2 768 2 384 2 192 2 96 2 48 2 24 2 12 2 6 3 3 1

768 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3

Here, prime factor 3 does not have its pair. If 3 gets a pair, then the number will become a perfect square. Therefore, 768 has to be multiplied with 3 to obtain a perfect square.

Therefore, 768 × 3 = 2304 is a perfect square.

768 × 3 = 2304 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3

∴ √2304 = 2 x 2 x 2 x 2 x 3 = 48

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