Get it on Google Play
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

Answer :

(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

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