question_answer 16)

                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                                   (iii) 1008                               (iv) 2028 (v) 1458                                (vi) 768

Answer:

                (i) 252 The prime factorization of 252 is \[252=2\times 2\times 3\times 3\times 7\]. As the prime factor 7 has no pair, 252 is not a perfect square. If 7 gets a pair, then the number will be a perfect square. So, we multiply 252 by 7 to get

2	252
2	126
3	63
3	21
	7

  \[252\times 7=\underline{2\times 2}\times \underline{3\times 3}\times \underline{7\times 7}\] Now each prime factor has a pair. Therefore, \[252\times 7=1764\] is a perfect square. Thus the required smallest number is 7. Thus, \[\sqrt{1764}=2\times 3\times 7=42\]. (ii) 180 The prime factorization of 180 is\[180=2\times 2\times 3\times 3\times 5\]. As the prime factor 5 has no pair, 180 is not a perfect square. If 5 gets a pair, then the number will be a perfect square. So, we multiply 180 by 5 to get

2	180
2	90
3	45
3	15
	5

                \[180\times 5=\underline{2\times 2}\times \underline{3\times 3}\times \underline{5\times 5}\]. Now each prime factor has a pair. Therefore, \[180\times 5=900\] is a perfect square. Thus the required smallest number is 5. Thus, \[\sqrt{900}\,=2\times 3\times 5=30\] (iii) 1008 The prime factorization of 1008 is\[1008=2\times 2\times 2\times 2\times 3\times 3\times 7\]. As the prime factor 7 has no pair, 1008 is not a perfect square. If 7 gets a pair, then the number will be a perfect square. So, we multiply 1008 by 7 to get

2	1008
2	504
2	252
2	126
3	63
3	21
	7

  \[1008\times 7=\underline{2\times 2}\,\times \underline{2\times 2}\times \underline{3\times 3}\times \underline{7\times 7}\] Now each prime factor has a pair. Therefore, 1008 x 7 = 7056 is a perfect square. Thus the required smallest number is 7. Thus,\[\sqrt{7056}=2\times 2\times 3\times 7=84\]. (iv) 2028 The prime factorisation of 2028 is \[2028=2\times 2\times 3\times 13\times 13\]. As the prime factor 3 has no pair, 2028 is not a perfect square. If 3 gets a pair, then the number will be a perfect square. So, we multiply 2028 by 3 to get

2	2028
2	1014
3	507
13	169
	13

\[2028\times 3=\underline{2\times 2}\times \underline{3\times 3}\times \underline{13\times 13}\] Now each prime factor has a pair. Therefore, \[2028\times 3=6084\] is a perfect square. Thus the required smallest number is 3. Thus, \[\sqrt{6084\text{ }}=2\times 3\times 13=78\]. (v) 1458 The prime factorization of 1458 is\[1458=2\times 3\times 3\times 3\times 3\times 3\times 3\]. As the prime factor 2 has no pair, 1458 is not a perfect square. If 2 gets a pair, then the number will be a perfect square. So, we multiply 1458 by 2 to get

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

  \[1458\,\times 2=\underline{2\times 2}\times \underline{3\times 3}\times \underline{3\times 3}\ \times \underline{3\times 3}\] Now each prime factor has a pair. Therefore, \[1458\times 2=2916\] is a perfect square. Thus the required smallest number is 2. Thus, \[\sqrt{2916}=2\times 3\times 3\times 3=54\]. (vi) 768 The prime factorisation of 768 is \[768=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 3\]. As the prime factor 3 has no pair, 768 is not a perfect square. If 3 gets a pair, then the number will be a perfect square. So, we multiply 768 by 3 to get

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

\[768\times 3=\underline{2\times 2}\times \underline{2\times 2}\times \underline{2\times 2}\times \underline{2\times 2}\times \underline{3\times 3}\] Now each prime factor has a pair. Therefore, \[768\times 3=2304\] is a perfect square. Thus the required smallest number is 3. Thus, \[\sqrt{2304}=48\].