Question

Using the Prime factorisation method , Find which of the following numbers are perfect squares .

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1. 768
2. 400
3. 630
4. 9025
5. 784
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Answer 1

Answer:

Hi for this you need to find the answer in such a way that they both are same and they together product the above any number

Answer 2

$\gray{ \underbrace{ \textbf{ \red{ \textsf{Question : }}}}}$

Using the Prime factorisation method , Find which of the following numbers are perfect squares.

1. 768

2. 400

3. 630

4. 9025

5. 784

$\gray{ \underbrace{ \textbf{ \green{ \textsf{Solution : }}}}}$

Q.1. 768

Using the prime factorisation

$\sf{ \underline{ 2\mid{768}}} \\ \sf { \underline{ 2\mid{384}}} \\ \sf{ \underline{ 2\mid{192}}} \\ \sf{ \underline{ 2\mid{96}}} \: \: \\ \sf{ \underline{ 2\mid{48 \: \: }}} \\ \sf{ \underline{ 2\mid{24 \: \: }}} \\ \sf{ \underline{ 2\mid{12 \: \: }}} \\ \sf{ \underline{ 2\mid{6 \: \: \: \: }}} \\ \sf{ \underline{ 3\mid{3 \: \: \: \: }}} \\ \sf{{ {1}}}$

After taking the L. C. M. of 768 prime factors we get :

$\\ \sf { \underline{2 \times 2 }} \: \times { \underline{ 2 \times 2}} \times { \underline{ 2 \times 2}} \times{ \underline{ 2 \times 2}} \times \bf 3$

➠ Prime factors = 2 × 2 × 2 × 2 √ 3

➠ Prime factors = 16√3

${ \underline{ \boxed{ \red{ \bf{Hence, \: it \: is \: not \: a \: perfect \: square. ❌ }}}}}$

Q.2. 400

Using prime factorisation method :

$\sf{ \underline{2 \mid{400}}} \\ \sf{ \underline{2 \mid{200}}} \\ \sf{ \underline{2 \mid{100}}} \\ \sf{ \underline{2 \mid{50 \: \: }}} \\ \sf{ \underline{5\mid{25 \: \: }}} \\ \sf{ \underline{5 \mid{5 \: \: \: \: }}} \\ \sf \: \: 1$

After taking the L. C. M. of 768 prime factors we get :

➠ $\\ \sf { \underline{2 \times 2 }} \: \times { \underline{ 2 \times 2}} \times { \underline{ 5 \times 5}}$

➠ Prime factors = 2 × 2 × 5

➠ Prime factors = 20

${ \underline{ \boxed{ \green{ \bf{Hence \: it \: is \: a \: perfect \: square ☑ }}}}}$

Q.3. 630

Using prime factorisation method :

$\sf{ \underline{ 2\mid{630}}} \\ \sf{ \underline{ 3\mid{315}}} \\ \sf{ \underline{ 3\mid{105 }}} \\ \sf{ \underline{ 5\mid{35 \: \: }}} \\ \sf{ \underline{ 7\mid{7 \: \: \: \: }}} \\ \: \sf 1$

Prime factors we get :

$\sf \to \bf 2 \times \sf{ \underline{3 \times 3}} \times \bf 5 \times 7 \\ \sf \to \sqrt[3]{2 \times 5 \times 7}$

➠ 3√2 × 5 × 7 can't be the perfect for √630.

${ \underline{ \boxed{ \red{ \bf{Hence \: it \: is \: not \: a \: perfect \: square ❌}}}}}$

Q.4. 9025

Using prime factorisation method :

$\sf{ \underline{ 5\mid{9025}}} \\ \sf{ \underline{ 5\mid{1085}}} \\ \sf{ \underline{ 19\mid{361 \: \: \: \: }}} \\ \sf{ \underline{19 \mid{19 \: \: \: \: \: \: }}} \\ 1$

Prime factors we get :

$\sf{ \underline{5 \times 5}} \times { \underline{19 \times 19}}$

➠ 19 × 5

➠ 95 will be the square of 9025.

${ \underline{ \boxed{ \green{ \bf{hence \: it \: is \: a \: perfect \: square ☑ }}}}}$

Q.5. 784

Using prime factorisation method :

$\sf{ \underline{ 2\mid{784}}} \\ \sf{ \underline{ 2\mid{392}}} \\ \sf{ \underline{ 2\mid{196}}} \\ \sf{ \underline{ 2\mid{98 \:\: \: }}} \\ \sf{ \underline{ 7\mid{49 \: \: }}} \\ \sf{ \underline{ 1\mid{7 \: \:\: \: }}} \\ \sf \: 1$

Prime factors we get :

➠ $\sf{ \underline{2 \times 2}} \times{ \underline{2 \times 2}} \times { \underline{7 \times 7}}$

➠ 2 × 2 × 7

➠ 28 can be the perfect no. for √784

${ \underline{ \boxed{ \green{ \bf{Hence \: it \: is \: a \: perfect \: square ☑ }}}}}$

$\green{ \underbrace{ \textbf{ \textsf{Required \:answer : }}}} \\ { \underline{ \boxed{ \textbf{ \textsf{Perfect \: squares \:no. are = \:400,\: 9025,\: 784 ☑}}}}}$