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(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

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

- 1.What could be the possible ‘one’s’ digits of the square root of each of the following numbers? (i) 9801 (ii) 99856 (iii) 998001 (iv) 657666025
- 2.Without doing any calculation, find the numbers which are surely not perfect squares. (i) 153 (ii) 257 (iii) 408 (iv) 441
- 3.Find the square roots of 100 and 169 by the method of repeated subtraction.
- 4.Find the square roots of the following numbers by the prime factorisation Method. (i) 729 (ii) 400 (iii) 1764 (iv) 4096 (v) 7744 (vi) 9604 (vii) 5929 (viii) 9216 (ix) 529 (x) 8100
- 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
- 7.The students of class VIII of a school donated ₹ 2401 in all, for the Prime Minister’s National Relief Fund. Each student donated as many rupees as the number of students in the class. Find the number of students in the class.
- 8.2025 plants are to be planted in a garden in such a way that each row contains as many plants as the number of rows. Find the number of rows and the number of plants in each row.
- 9.Find the smallest square number that is divisible by each of the numbers 4, 9 and 10.
- 10.Find the smallest number that is divisible by each of the numbers 8, 15 and 20.

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