Foreach of the following numbers, find the smallest whole number by which its should be multiplied so as to get a perfect square number

(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 apair, then the number will become aperfect square.

There fore, 252 has to be  multipliedwith 7 to obtain aperfect square

252 × 7 = 2 × 2 × 3 × 3 × 7 × 7

Hence,

252 × 7 = 1764 is a perfect square

√1764 = 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 apair, then the number will become aperfect square.

There fore, 180 has to be multiplied with 5 to obtain a perfect square.

180 × 5 = 900 = 2 ×  2 × 3 × 3 × 5 × 5

There fore,

180 × 5 = 900 is a perfect square

√900 = 2*3*5

= 30

(iii)The prime factorization of 1008 is as follows:

1008 = 2 × 2 × 2 × 2 × 3 × 3 × 7

Here,primefactor 7 does not have its pair.

If 7 gets apair, then the number will become aperfect square.

There fore,1008 can be multiplied with 7 to obtain aperfect square.

1008 × 7 = 7056 = 2 × 2 × 2 × 2 × 3 × 3 × 7 × 7

Therefore,

1008 × 7 = 7056 is a perfect square

√7056 = 2 * 2 * 3 * 7

= 84

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

2028 = 2 * 2 * 3 * 13 * 13

Here,prime factor 3 doesnot have its pair . If 3 gets apair, then the number will become aperfect square. There fore, 2028 can be multiplied with 3 to obtain aperfect square.

2028 × 3 = 6084 = 2 × 2 × 3 × 3 × 13 × 13

There fore,

2028 × 3 = 6084 is a perfect square

√6084 = 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 apair, then the number will be come a perfect square.

Therefore, 1458 can be multiplie dwith 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

√2916 = 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 d oes not have its pair.

If 3 gets apair, then the number will be comeaper fect square.

There fore, 768 can be multiplied with 3 to obtain aperfect square.

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

Therefore,

768×3=2304isaperfectsquare

√2304 = 2 * 2 * 2 * 2 * 3

= 48