Question 1: Find the square of each of the following numbers:

(i) $14 = 14$     (ii) $137$     (iii) $\frac{4}{17}$     (iv) $2$ $\frac{3}{4}$     (v) $0.01$     (vi) of $1.2$     (vii) $0.17$
(viii) $4.6$

(i) Square of $14 = 14 \times 14 = 196$

(ii) Square of $137 = 137 \times 137 = 18769$

(iii) Square of $\displaystyle \frac{4}{17} =\frac{16}{289}$

(iv) Square of $\displaystyle 2 \frac{3}{4} = \frac{121}{16}$

(v) Square $0.01 = 0.01 \times 0.01 = 0.0001$

(vi) Square of $1.2 = 1.2 \times 1.2 = 1.44$

(vii) Square of $0.17 =0.17 \times 0.17 = 0.0289$

(viii) Square of $4.6 = 4.6 \times 4.6 = 21.16$

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Question 2: Using prime factorization method, find which of the following are perfect square numbers:

(i) $196$     (ii) $252$     (iii) $324$     (iv) $1225$     (v) $2916$     (vi) $3582$     (vii) $4489$

Note: A natural number is called a perfect square, if it is the square of some natural number

(i) $196 = 14 \times 14$ (hence perfect square)

(ii) $252 = 2 \times 2 \times 7 \times 3 \times 3$ (not a perfect square)

(iii) $324 = 18 \times 18$ (hence perfect square)

(iv) $1225 = 35 \times 35$ (hence a perfect square)

(v) $2916 = 54 \times 54$ (hence perfect square)

(vi) $3582 = 2 \times 3 \times 3 \times 199$ (not a perfect square)

(vii) $4489 = 67 \times 67$ (hence a perfect square)

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Question 3: Which of the following numbers are squares of even numbers?
Note: The Square of an even number is always an even number.

$676$,          $1089$,           $5625$,           $729$,           $2304$,           $9216$

$676$ (square of $26$ ) $2304$ (square of $48$ ) and $9216$ (square of $96$) are square of even numbers.

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Question 4: Using prime factorization method, find the square root of each of the following numbers:

(i) $441$     (ii) $784$     (iii) $3969$     (vi) $4900$    (v) $11025$    (vi) $30625$

(i) $441 = 3 \times 7 \times 3 \times 7$. $\therefore$, square root of $441 = 3 \times 7 = 21$

(ii) $784 = 4 \times 7 \times 4 \times 7$. Therefore, square root of $784 = 4 \times 7 = 28$

(iii) $3969 = 7 \times 9 \times 7 \times 9$. Therefore, square root of $3969 = 7 \times 9 = 63$

(vi) $4900 = 7 \times 10 \times 7 \times 10$. Therefore, square root of $4900 = 70$

(v) $11025 = 3 \times 7 \times 5 \times 3 \times 7 \times 5$. Therefore, square root of $11025 = 3 \times 7 \times 5 = 105$

(vi) $30625 = 5 \times 5 \times 7 \times 5 \times 5 \times 7$. Therefore, square root of $30625 = 5 \times 7 \times 5 = 175$

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Question 5: The students of a class arranged a picnic. Each student contributed as many rupees as the number of students in the class. If the total contribution is Rs. $2601$ , find the strength of the class.

Let the number of students $= x$

Each student contributed $x$ Rupees.

Therefore $x^2 = 2601$ or $x = 51$

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Question 6: Find the smallest number by which $588$ be multiplied to get a perfect square number.

$588 = 2 \times 2 \times 7 \times 3 \times 7$. Therefore multiply by $3$ to get a perfect square

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Question 7: Find the smallest number by which $2400$ be multiplied to get a perfect square number. Find the square root of the resulting number.

$2400 = 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 5 \times 5$. Therefore multiply by $6$. The square root would be $120$

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Question 8: Find the smallest number by which $2592$ be multiplied to get a perfect square number.

(i) What is the perfect square number so obtained?

$2592 = 2 \times 2 \times 2 \times 2 \times 2 \times 9 \times 9$.

Therefore smallest number to be multiplied to $2592$ to get a perfect square is $2$. Perfect square number $= 5184$

(ii) What is the square root of the resulting number?

Square root of the resulting number is $72$

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Question 9: Find the smallest number by which $1728$ be divided to get a perfect square number.

(i) What is the perfect square number so obtained?

$1728 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3$. Hence divide it by $3$. The number would be $576$

(ii) Find the square root of this number.

Square root $= 24$

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Question 10: Find the smallest number by which 7776 be divided to get a perfect square number.

(i) What is the resulting number?    (ii) What is the square root of the number so obtained?

(i) What is the resulting number?

$7776 = 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 9 \times 9$. Hence divide this by $6$. The number would be $1296$

(ii) What is the square root of the number so obtained?

Square root $= 36$

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Question 11: Find the least square number which is exactly divisible by each of the numbers $8, 9, 10$ and $15$.

$8 = 2 \times 2 \times 2$

$9 = 3 \times 3$

$10 = 2 \times 5$

$15 = 3 \times 5$

Therefore the number is $2 \times 2 \times 2 \times 5 \times 3 \times 3 = 360$

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Question 12: Find the square root of each of the following by division method:

i) $961$    ii) $5476$    iii) $11449$    iv) $225625$    v) $4401604$    vi) $9653449$

By the long division method, we have

i) $961$

${3 \hspace{0.5cm}} | \overline{9} \hspace{0.5cm} \overline{61} \hspace{0.5cm} \overline{} \hspace{0.5cm} ( 31 \\ {\hspace{0.7cm}| } \underline{ 9 \hspace{2.0cm} } \\ 61 {\hspace{0.28cm}| } {\hspace{0.2cm} } \hspace{0.5cm} 61 \\ {\hspace{0.7cm}| } \underline{ {\hspace{0.2cm} } \hspace{0.5cm} 61 } \\ {\hspace{0.7cm}| } {\hspace{0.2cm} } \hspace{0.4cm} 0 \hspace{0.5cm}$

ii) $5476$

${7 \hspace{0.7cm}} | \overline{54} \hspace{0.5cm} \overline{76} \hspace{0.5cm} \overline{} \hspace{0.5cm} ( 74 \\ {\hspace{0.9cm}| } \underline{ 49 \hspace{2.0cm} } \\ 144 {\hspace{0.28cm}| } {\hspace{0.2cm} 5} \hspace{0.5cm} 76 \\ {\hspace{0.9cm}| } \underline{ {\hspace{0.2cm} 5} \hspace{0.5cm} 76 } \\ {\hspace{0.9cm}| } {\hspace{0.2cm} } \hspace{0.4cm} 0 \hspace{0.5cm}$

iii) $11449$

${1 \hspace{0.7cm}} | \overline{1} \hspace{0.5cm} \overline{14} \hspace{0.5cm} \overline{49} \hspace{0.5cm} \overline{} \hspace{0.5cm} ( 107 \\ {\hspace{0.9cm}| } \underline{ 1 \hspace{3.0cm} } \\ 207 {\hspace{0.28cm}| } {\hspace{0.7cm} 14} \hspace{0.5cm} 49 \\ {\hspace{0.9cm}| } \underline{ {\hspace{0.7cm} 14} \hspace{0.5cm} 49 \hspace{1.5cm} } \\ {\hspace{0.9cm}| } {\hspace{0.7cm} } \hspace{0.7cm} 0 \hspace{0.5cm}$

iv) $225625$

${4 \hspace{0.7cm}} | \overline{22} \hspace{0.5cm} \overline{56} \hspace{0.5cm} \overline{25} \hspace{0.5cm} \overline{} \hspace{0.5cm} ( 475 \\ {\hspace{0.9cm}| } \underline{ 16 \hspace{3.0cm} } \\ 87 {\hspace{0.48cm}| } {\hspace{0.3cm} 6} \hspace{0.5cm} 56 \\ {\hspace{0.9cm}| } \underline{ {\hspace{0.3cm} 6} \hspace{0.5cm} 09 \hspace{1.5cm} } \\ 945 {\hspace{0.28cm}| } {\hspace{0.95cm} 47} \hspace{0.5cm} 25 \hspace{0.5cm} \\ {\hspace{0.9cm}| } \underline{ {\hspace{0.95cm} 47} \hspace{0.5cm} 25 \hspace{0.5cm} } \\ {\hspace{0.9cm}| } {\hspace{0.7cm} } \hspace{0.7cm} 0 \hspace{0.5cm}$

v) $4401604$

${4 \hspace{1.0cm}} | \overline{4} \hspace{0.5cm} \overline{40} \hspace{0.5cm} \overline{16} \hspace{0.5cm} \overline{04} \hspace{0.5cm} ( 2098 \\ {\hspace{1.2cm}| } \underline{ 4 \hspace{3.0cm} } \\ 409 {\hspace{0.58cm}| } {\hspace{0.7cm} 40} \hspace{0.5cm} 16 \\ {\hspace{1.2cm}| } \underline{ {\hspace{0.7cm} 36} \hspace{0.5cm} 81 \hspace{1.5cm} } \\ 4188 {\hspace{0.38cm}| } {\hspace{0.7cm} 3} \hspace{0.5cm} 35 \hspace{0.5cm} 04 \\ {\hspace{1.2cm}| } \underline{ {\hspace{0.7cm} 3} \hspace{0.5cm} 35 \hspace{0.5cm} 04 } \\ {\hspace{1.2cm}| } {\hspace{0.7cm} } \hspace{0.7cm} 0 \hspace{0.5cm}$

vi) $9653449$

${3 \hspace{1.0cm}} | \overline{9} \hspace{0.5cm} \overline{65} \hspace{0.5cm} \overline{34} \hspace{0.5cm} \overline{49} \hspace{0.5cm} ( 3107 \\ {\hspace{1.2cm}| } \underline{ 9 \hspace{3.0cm} } \\ 61 {\hspace{0.78cm}| } {\hspace{0.7cm} 65} \hspace{0.5cm} \\ {\hspace{1.2cm}| } \underline{ {\hspace{0.7cm} 61} \hspace{0.5cm} \hspace{1.5cm} } \\ 6207 {\hspace{0.38cm}| } {\hspace{0.7cm} 3} \hspace{0.5cm} 34 \hspace{0.5cm} 49 \\ {\hspace{1.2cm}| } \underline{ {\hspace{0.7cm} 4} \hspace{0.5cm} 34 \hspace{0.5cm} 49 } \\ {\hspace{1.2cm}| } {\hspace{0.7cm} } \hspace{0.7cm} 0 \hspace{0.5cm}$

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Question 13: The area of a square field is $77841$ sq. meters. Find its perimeter.

Area = side $\times$ side $= 77841 = 279$ meter

Perimeter $= 4$ side $= 4 279 = 1116$ sq. meters

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Question 14: Find the least number which must be subtracted from $7581$ to obtain a perfect square. Find this perfect square and its square root.

${8 \hspace{0.5cm}} | \overline{75} \hspace{0.5cm} \overline{81} \hspace{0.5cm} \overline{} \hspace{0.5cm} ( 31 \\ {\hspace{0.7cm}| } \underline{ 64 \hspace{2.0cm} } \\ 167 {\hspace{0.08cm}| } {\hspace{0.2cm} 11} \hspace{0.5cm} 81 \\ {\hspace{0.7cm}| } \underline{ {\hspace{0.2cm} 11} \hspace{0.5cm} 69 } \\ {\hspace{0.7cm}| } {\hspace{0.2cm} } \hspace{0.4cm} 12 \hspace{0.5cm}$

Subtract $12$ from $7581$ to obtain a perfect square. The number would be $7569$ and the square root would be $87$.

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Question 15: Find the least number which must be subtracted from $43379$ to obtain a perfect Find this perfect square and its square root.

${2 \hspace{0.7cm}} | \overline{4} \hspace{0.5cm} \overline{33} \hspace{0.5cm} \overline{79} \hspace{0.5cm} \overline{} \hspace{0.5cm} ( 208 \\ {\hspace{0.9cm}| } \underline{ 4 \hspace{3.0cm} } \\ 207 {\hspace{0.28cm}| } {\hspace{0.7cm} 33} \hspace{0.5cm} 79 \\ {\hspace{0.9cm}| } \underline{ {\hspace{0.7cm} 32} \hspace{0.5cm} 64 \hspace{1.5cm} } \\ {\hspace{0.9cm}| } {\hspace{0.7cm} 1} \hspace{0.7cm} 15 \hspace{0.5cm}$

Subtract $115$ from $43379$ to obtain perfect square.

Question 16: Find the least number which must be added to 6203 to obtain a perfect square. Find the perfect square and its square

${7 \hspace{0.7cm}} | \overline{62} \hspace{0.5cm} \overline{03} \hspace{0.5cm} \overline{} \hspace{0.5cm} ( 78 \\ {\hspace{0.9cm}| } \underline{ 49 \hspace{2.0cm} } \\ 148 {\hspace{0.28cm}| } {\hspace{0.1cm} 13} \hspace{0.5cm} 03 \\ {\hspace{0.9cm}| } \underline{ {\hspace{0.1cm} 11} \hspace{0.5cm} 84 } \\ {\hspace{0.9cm}| } {\hspace{0.2cm} 1 } \hspace{0.4cm} 19 \hspace{0.5cm}$

Therefore $78^2 < 7203 < 79^2$

$79^2 = 6241$

Therefore add $(6241-6203) = 38$ to $6203$ to obtain a perfect square $(6241)$.

Its square root would be $79$.

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Question 17: Find the least number which must be added to 506900 to make it a perfect square. Find this perfect square and its square root.

${7 \hspace{0.7cm}} | \overline{50} \hspace{0.5cm} \overline{69} \hspace{0.5cm} \overline{00} \hspace{0.5cm} \overline{} \hspace{0.5cm} ( 711 \\ {\hspace{0.9cm}| } \underline{ 49 \hspace{3.0cm} } \\ 141 {\hspace{0.28cm}| } {\hspace{0.3cm} 1} \hspace{0.5cm} 69 \\ {\hspace{0.9cm}| } \underline{ {\hspace{0.3cm} 1} \hspace{0.5cm} 41 \hspace{1.5cm} } \\ 1421 {\hspace{0.08cm}| } {\hspace{0.95cm} 28} \hspace{0.5cm} 00 \hspace{0.5cm} \\ {\hspace{0.9cm}| } \underline{ {\hspace{0.95cm} 14} \hspace{0.5cm} 28 \hspace{0.5cm} }$

Therefore $711^2 < 506900 < 712^2$

$712^2 = 506944$

Therefore add $44$ to $506900$ to make it a perfect square of $712$.

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Question 18:  Find the greatest number of six digits, which is a perfect square. Find the square root of this number.

${9 \hspace{1.0cm}} | \overline{99} \hspace{0.5cm} \overline{99} \hspace{0.5cm} \overline{99} \hspace{0.5cm} \overline{} \hspace{0.5cm} ( 999 \\ {\hspace{1.2cm}| } \underline{ 81 \hspace{3.0cm} } \\ 189 {\hspace{0.55cm}| } {\hspace{0.1cm} 18} \hspace{0.5cm} 99 \\ {\hspace{1.2cm}| } \underline{ {\hspace{0.1cm} 17} \hspace{0.5cm} 01 \hspace{1.5cm} } \\ 1989 {\hspace{0.38cm}| } {\hspace{0.1cm} 1} \hspace{0.5cm} 98 \hspace{0.5cm} 99 \\ {\hspace{1.2cm}| } \underline{ {\hspace{0.1cm} 1} \hspace{0.5cm} 79 \hspace{0.5cm} 01 } \\ {\hspace{1.2cm}| } {\hspace{0.7cm} 19} \hspace{0.7cm} 98 \hspace{0.5cm}$

Subtract $1998$ from $999999$ to make a perfect square. The number is $998001$.

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Question 19: Find the least number of four digits which is a perfect square.

${3 \hspace{0.7cm}} | \overline{10} \hspace{0.5cm} \overline{00} \hspace{0.5cm} \overline{} \hspace{0.5cm} ( 31 \\ {\hspace{0.9cm}| } \underline{ 9 \hspace{2.0cm} } \\ 61 {\hspace{0.48cm}| } {\hspace{0.1cm} 1} \hspace{0.5cm} 00 \\ {\hspace{0.9cm}| } \underline{ {\hspace{0.1cm} } \hspace{0.7cm} 61 } \\ {\hspace{0.9cm}| } {\hspace{0.2cm} } \hspace{0.5cm} 39 \hspace{0.5cm}$

Therefore $31^2 < 1000 < 32^2$

$32^2 = 1024$

Therefore add $24$ to $1000$ to get the least number of four digits which is a perfect square which is $1024$.

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Question 20: Find the least number by which $69192$ must be (i) decreased (ii) increased (iii) multiplied (iv) divided to make it a perfect.

${2 \hspace{0.7cm}} | \overline{6} \hspace{0.5cm} \overline{91} \hspace{0.5cm} \overline{92} \hspace{0.5cm} \overline{} \hspace{0.5cm} ( 263 \\ {\hspace{0.9cm}| } \underline{ 4 \hspace{3.0cm} } \\ 46 {\hspace{0.48cm}| } {\hspace{0.3cm} 2} \hspace{0.5cm} 91 \\ {\hspace{0.9cm}| } \underline{ {\hspace{0.3cm} 2} \hspace{0.5cm} 76 \hspace{1.5cm} } \\ 52 {\hspace{0.48cm}| } {\hspace{0.95cm} 15} \hspace{0.5cm} 92 \hspace{0.5cm} \\ {\hspace{0.9cm}| } \underline{ {\hspace{0.95cm} 15} \hspace{0.5cm} 69 \hspace{0.5cm} } \\ {\hspace{0.9cm}| } {\hspace{0.7cm} } \hspace{0.7cm} 23 \hspace{0.5cm}$
Subtract $23$ from $69192$ to make it a perfect square.
$263^2 < 69191 < 2642, 264^2 = 69696$. Therefore add $504$ to $69192$ to make it a perfect square.
$69192 = 2 \times 2 \times 2 \times 3 \times 3 \times 31 \times 31$. Therefore multiply by $2$ to make it a perfect square
Or divide it by $2$ to make it a perfect square.