Question

find the smallest number by which 6192 must be multiplied so that product is a perfect square.​

Answer 1

$\large\underline{\sf{Solution-}}$

Let first find the prime factorization of 6192.

$\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:6192 \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:3096 \:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:1548\:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:774 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:387 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:129 \:\:}} \\ {\underline{\sf{43}}}& \underline{\sf{\:\:43 \:\:}} \\ \underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}$

Hence,

Prime factorization of 6192 is

$\rm :\longmapsto\:6192 = \underbrace{2 \times 2} \times \underbrace{2 \times 2} \times \underbrace{3 \times 3} \times 43$

In order to make it a perfect square, we need to complete the pairs.

So, to make 6192 a perfect square, it should be multiplied by 43.

So, required number is 6192 × 43 = 266256

and

$\rm :\longmapsto\: \sqrt{266256} = 516$

Additional Information :-

Question : -

Find the smallest number by which 6336 must be divided so that product is a perfect square.

Answer :-

Let us first find the prime factorization of 6336.

$\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:6336 \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:3168 \:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:1584\:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:792 \:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:396 \:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:198 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:99 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:33 \:\:}} \\ {\underline{\sf{11}}}& \underline{\sf{\:\:11 \:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}$

Hence,

Prime factorization of 6336 is

$\rm :\longmapsto\:6336 = \underbrace{2 \times 2} \times \underbrace{2 \times 2} \times\underbrace{2 \times 2} \times \underbrace{3 \times 3} \times 43$

In order to make it a perfect square, we need to complete the pairs.

So, to make 6336 a perfect square, it should be divided by 11.

So, required number is 6336 ÷ 11 = 576

and

$\rm :\longmapsto\: \sqrt{576} = 24$

Answer 2

\large\underline{\sf{Solution-}}

Solution−

Let first find the prime factorization of 6192.

\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:6192 \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:3096 \:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:1548\:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:774 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:387 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:129 \:\:}} \\ {\underline{\sf{43}}}& \underline{\sf{\:\:43 \:\:}} \\ \underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}\end{gathered}

2

2

2

2

3

3

43

6192

3096

1548

774

387

129

43

1

Hence,

Prime factorization of 6192 is

\rm :\longmapsto\:6192 = \underbrace{2 \times 2} \times \underbrace{2 \times 2} \times \underbrace{3 \times 3} \times 43:⟼6192=

2×2

×

2×2

×

3×3

×43

In order to make it a perfect square, we need to complete the pairs.

So, to make 6192 a perfect square, it should be multiplied by 43.

So, required number is 6192 × 43 = 266256

and

\rm :\longmapsto\: \sqrt{266256} = 516:⟼

266256

=516

Additional Information :-

Question : -

Find the smallest number by which 6336 must be divided so that product is a perfect square.

Answer :-

Let us first find the prime factorization of 6336.

\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:6336 \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:3168 \:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:1584\:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:792 \:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:396 \:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:198 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:99 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:33 \:\:}} \\ {\underline{\sf{11}}}& \underline{\sf{\:\:11 \:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}\end{gathered}

2

2

2

2

2

2

3

3

11

6336

3168

1584

792

396

198

99

33

11

1

Hence,

Prime factorization of 6336 is

\rm :\longmapsto\:6336 = \underbrace{2 \times 2} \times \underbrace{2 \times 2} \times\underbrace{2 \times 2} \times \underbrace{3 \times 3} \times 43:⟼6336=

2×2

×

2×2

×

2×2

×

3×3

×43

In order to make it a perfect square, we need to complete the pairs.

So, to make 6336 a perfect square, it should be divided by 11.

So, required number is 6336 ÷ 11 = 576

and

\rm :\longmapsto\: \sqrt{576} = 24:⟼

576

=24