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 $2304 = 2 \times 2 \times 2 \times 2 \times 3 = 48$