Question

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. 

1. 252 
2.  180 
3.  1008 
4.  2028 
5.  1458 
6.  768

Answer 1

(i) \(252\) can be factorised as follows.

2	252
2	126
3	63
3	21
7	7
	1

\(252 = \underline{2 × 2} × \underline{3 × 3} × 7\)
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.
\(252 × 7\) = \(\underline{2 × 2} × \underline{3 × 3} × \underline{7 × 7}\)
Therefore, \(252 × 7 = 1764\) is a perfect square. 
∴ \(\sqrt{1764} = 2 \times 3 \times 7 = 42\)

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(ii)\(180\) can be factorised as follows.

2	180
2	90
3	45
3	15
5	5
	1

 \(180 = \underline{2 × 2} × \underline{3 × 3 }× 5\)
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.
\(180 × 5 = 900 = \underline{2 × 2} × \underline{3 × 3} × \underline{5 × 5}\)
Therefore, \(180 × 5 = 900 \) is a perfect square. 
∴ \(\sqrt{900} = 2\times3\times5=30\)

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(iii)\(1008\) can be factorised as follows. 

2	1008
2	504
2	250
2	126
3	63
3	21
7	7
	1

\(1008 = \underline{2 × 2} × \underline{2 × 2} × \underline{3 × 3} × 7\)
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.
\(1008 × 7 = 7056 = \underline{2 × 2} ×\underline{2 × 2} × \underline{3 × 3} × \underline{7 × 7}\)
Therefore, \( 1008 × 7 = 7056\) is a perfect square
∴ \(\sqrt{7056}=2\times2*3\times7=84\)

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(iv) 2028 can be factorised as follows. 

2	2028
2	1014
3	507
13	169
13	13
	1

\(2028 = \underline{2 \times 2} \times 3 \times \underline{13 \times 13 }\) 
Here, prime factor \(3\) has no pair. 
Therefore \(2028\) must be multiplied by \(3\) to make it a perfect square.  
\(2028 \times 3 = 6084 \;  And \;\sqrt{6084} = 2 \times 2 \times 3 \times 3 \times 13 \times 13 = 78\)

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(v) \(1458 = 2 \times \underline{3 \times 3} \times \underline{3 \times 3} \times \underline{3 \times 3}\)  

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

Here, prime factor \(2\) has no pair. 
Therefore \(1458 \) must be multiplied by \(2\) to make it a perfect square.  
\(\therefore\) \(1458 \times 2 = 2916\)   And \(2916 = 2 \times 3 \times 3 \times 3 = 54\)

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(vi) \(768 = \underline{2 \times 2} \times \underline{2 \times 2} \times \underline{2 \times 2 }\times \underline{2 \times 2 }\times 3 \)

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

 Here, prime factor \(3\) has no pair. 
Therefore \(768\) must be multiplied by \(3\) to make it a perfect square.  
\(768 \times 3 = 2304 \)  And \(2304 = 2 \times 2 \times 2 \times 2 \times 3 = 48\)