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

(i) 252 (ii) 180

(iii) 1008 (iv) 2028

(v) 1458 (vi) 768

(i)The prime factorization of 252 is follows:

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__

Hence,

252 × 7 = 1764 is a perfect square

2 * 3 * 7

= 42

(ii) The prime factorization of 180 is as follows:

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

= 2 * 3 * 5

= 30

(iii) The prime factorization of 1008 isas follows:

1008 = __2 × 2__ × __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, 1008 can be multiplied with 7 to obtain a perfect square.

1008 × 7 = 7056 = __2 × 2__ ×__2 × 2__ × __3 × 3__ × __7 × 7__

Therefore,

1008 × 7 = 7056 is a perfect square

= 2 * 2 * 3 * 7

= 84

(iv) The prime factorization of 2028 is as follows:

2028 = __2 * 2__ * 3 * __13 * 13__

Here, prime factor 3 does not have its pair. If 3 gets a pair, then the number will become a perfect square. Therefore, 2028 can be multiplied with 3 to obtain a perfect square.

2028 × 3 = 6084 = __2 × 2__ ×__3 × 3__ × 1__3 × 13__

Therefore,

2028 × 3 = 6084 is a perfect square

= 2 * 3 * 13

= 78

(v) The prime factorization of 1458 is as follows:

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 can be multiplied with 2 to obtain a perfect square.

1458 × 2 = 2916 = __2 × 2__ ×__3 × 3__ × __3 × 3__ × __3 × 3__

Therefore,

1458 × 2 = 2916 is a perfect square

= 2 * 3 * 3 * 3

= 54

(vi) The prime factorization of 768 is as:

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 can be multiplied with 3 to obtain a perfect square.

768 × 3 = 2304 = __2 × 2__ ×__2 × 2__ × __2 × 2__ × __2 × 2__ ×__3 ×3__

Therefore,

768 × 3 = 2304 is a perfect square

= 2 * 2 * 2 * 2 * 3

= 48

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