# 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) 252  (ii) 180

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(i)252 can be factorised as follows.

 2 252 2 126 3 63 3 21 7 7 1

252 = 2 × 2 × 3 × 3 × 7

Here, prime factor 7 does not have its pair. If 7 gets a pair, then the number will become a perfect square. Therefore, 252 has to be multiplied with 7 to obtain a perfect square.

252 × 7 = 2 × 2 × 3 × 3 × 7 × 7

Therefore, 252 × 7 = 1764 is a perfect square

∴ √1764 = 2 x 3 x 7 = 42

(ii)180 can be factorised as follows.

 2 180 2 90 3 45 3 45 5 5 1

180 = 2 × 2 × 3 × 3 × 5

Here, prime factor 5 does not have its pair. If 5 gets a pair, then the number will become a perfect square. Therefore, 180 has to be multiplied with 5 to obtain a perfect square.

180 × 5 = 900 = 2 × 2 × 3 × 3 × 5 × 5

Therefore, 180 × 5 = 900 is a perfect square.

∴ √900 = 2 x 3 x 5 = 30