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


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 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 = \(1620 ÷ 5 = 324\)

∴ \(\sqrt{324} = 18\)

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