5.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

(i) We have prime factors of 252 as:
\(\qquad \begin{array}{c|lcr} 2 & 252\\ \hline 2 & 126\\ \hline 3 & 63\\ \hline 3 & 21\\ \hline 7 & 7\\ \hline & 1\\ \end{array} \)

\(252 = 2 \times 2 \times 3 \times 3 \times 7\)

Here, the prime factors are not in pair as 7 has no pair. Thus, the smallest whole number by which the given number is to be multiplied to get a perfect square number is 7.

The new square number is \(252 \times 7 = 1764\)

∴Square root of 1764 is

\(\sqrt{1764} = 2 \times 3 \times 7 = 42\)

(ii)We have prime factors of180 as:

\( \begin{array}{c|lcr} 2 & 180\\ \hline 2 & 90\\ \hline 5 & 45\\ \hline 3 & 9\\ \hline 3 & 3\\ \hline & 1\\ \end{array} \)

\(180 = 2 \times 2 \times 3 \times 3 \times 5\)

Here, 5 has no pair.So the least number to be multiplied to get perfect square is 5

New square number = \(180 \times 5 = 900\)

The square root of 900 is

∴\(\sqrt{900} = 2 \times 3 \times 5 = 30\)

(iii) We have prime factors of 1008 as:

\( \begin{array}{c|lcr} 2 & 1008\\ \hline 2 & 504\\ \hline 2 & 252\\ \hline 2 & 126\\ \hline 3 & 63\\ \hline 3 & 21\\ \hline 7 & 7\\ \hline & 1\\ \end{array} \)

\(1008 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 7\)

Here, 7 has no pair.So the least number to be multiplied to get perfect square is 7

New square number = \(1008 \times 7 = 7056\)

Square root of 7056 is

∴\(\sqrt{7056}= 2 \times 2 \times 3 \times 7 = 84\)

(iv) We have prime factors of 2028 as:

\( \begin{array}{c|lcr} 2 & 2028\\ \hline 2 & 1014\\ \hline 3& 507\\ \hline 13 & 169\\ \hline 13 & 13\\ \hline & 1\\ \end{array} \)

\(2028 = 2 \times 2 \times 3 \times 13 \times 13\)

Here, 3 is not in pair.So the least number to be multiplied to get perfect square is 3

New square number =\(2028 \times 3 = 6084\)

Square root of 6084 is

∴\(\sqrt{6084 }= 2 \times 13 \times 3 = 78\)

(v)We have prime factors of 1458 as:

\( \begin{array}{c|lcr} 2 & 1458\\ \hline 2 & 729\\ \hline 3 & 243\\ \hline 3 & 81\\ \hline 3 & 27\\ \hline 3 & 9\\ \hline 3 & 3\\ \hline & 1\\ \end{array} \)

\(1458 = 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3\)

Here, 2 is not in pair.So the least number to be multiplied to get perfect square is 2

New square number =\( 1458 \times 2 = 2916\)

Square root of 2916 is

∴\(\sqrt{2916 }= 3 \times 3 \times 3 \times 2 = 54\)

(vi) We have prime factors of 768 as:

\( \begin{array}{c|lcr} 2 & 768\\ \hline 2 & 384\\ \hline 2 & 192\\ \hline 2 & 96\\ \hline 2 & 48\\ \hline 2 & 24\\ \hline 2 & 12\\ \hline 2 & 6\\ \hline 3 & 3\\ \hline & 1\\ \end{array} \)

\(768 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3\)

Here, 3 is not in pair.So the least number to be multiplied to get perfect square is 3

New square number = \(768 \times 3 = 2304\)

Square root of 2304 is

∴\(\sqrt{2304} = 2 \times 2 \times 2 \times 2 \times 3 = 48\)