6.For each of the following numbers, find the smallest whole number by which it should be divided so as to get a perfect square. Also, find the square root of the square number so obtained.
(i) 252

(ii) 2925

(iii) 396

(iv) 2645

(v) 2800

(vi) 1620

(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 divided to get a perfect square number is 7.

The new square number is \(252 ÷ 7 = 36\)

∴Square root of 36 is \(\sqrt{36} = 6\)

(ii) We have prime factors of 2925 as:
\(\qquad \begin{array}{c|lcr} 3 & 2925\\ \hline 3 & 975\\ \hline 5 & 325\\ \hline 5 & 65\\ \hline 13 & 13\\ \hline & 1\\ \end{array} \)

\(2925 = 3 \times 3 \times 5 \times 5 \times 13\)

Here, 13 has no pair.So, the smallest whole number by which 2925 is divided to get a square number is 13.

New square number =\( 2925 ÷ 13 = 225\)

∴ \(\sqrt{225} = 15\)

(iii)We have prime factors of 396 as:

\(\qquad \begin{array}{c|lcr} 2 & 396\\ \hline 2 & 198\\ \hline 3 & 99\\ \hline 3 & 33\\ \hline 11 & 11\\ \hline & 1\\ \end{array} \)

\(396 = 2 \times 2 \times 3 \times 3 \times 11\)

Here 11 is not in pair.So,the required smallest whole number by which 396 is divided to get a square number is 11.

New square number = \(396 ÷ 11 = 36\)

∴ \(\sqrt{36} = 6\)

(iv)We have prime factors of 2645 as:

\(\qquad \begin{array}{c|lcr} 5 & 2645\\ \hline 23 & 529\\ \hline 23 & 23\\ \hline & 1\\ \end{array} \)

\(2645 = 5 \times 23 \times 23\)

Here, 5 is not in pair.So, 5 is the required smallest whole number by which 2645 is divided to get a square number

New square number = \(2645 ÷ 5 = 529\)

∴ \(\sqrt{529} = 23\)

(v) We have prime factors of 2800 as:

\(\qquad \begin{array}{c|lcr} 2 & 2800\\ \hline 2 & 1400\\ \hline 2 & 700\\ \hline 2 & 350\\ \hline 5 & 175\\ \hline 5 & 35\\ \hline 7 & 7\\ \hline & 1\\ \end{array} \)

\(2800 = 2 \times 2 \times 2 \times 2 \times 5 \times 5 \times 7\)

Here, 7 is not in pair.So, 7 is the required smallest number by which 2800 is divided to get a square number.

New square number =\( 2800 ÷ 7 = 400\)

∴ \(\sqrt{400} = 20\)

(vi)We have prime factors of 1620 as:

\(\qquad \begin{array}{c|lcr} 2 & 1620\\ \hline 2 & 810\\ \hline 3 & 405\\ \hline 3 & 135\\ \hline 3 & 45\\ \hline 3 & 15\\ \hline 5 & 5\\ \hline & 1\\ \end{array} \)

\(1620 = 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 5\)

Here, 5 is not in pair.So, 5 is the required smallest prime number by which 1620 is divided to get a square number = \(1620 ÷ 5 = 324\)

∴ \(\sqrt{324} = 18\)