Question

Find the smallest whole number by which 7128 should be divided to get a perfect square number. Also, find the square root of the number so obtained.

Answer 1

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

Given number is 7128

Let we first find the prime factorization of 7128

$\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:7128 \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:3564 \:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:1782\:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:891 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:297 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:99 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:33 \:\:}} \\ {\underline{\sf{11}}}& \underline{\sf{\:\:11 \:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}$

Now,we have prime factorization of 7128 as

$\rm :\longmapsto\:7128 =2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 2 \times 11$

Now, 7128 have two alone pairs of 2 and 11, so as to make it a perfect square, we have to divide 7128 by 22.

So required number is 7128 ÷ 22 = 324

Now,

Prime factorization of 324 is

$\rm :\longmapsto\:324 = 2 \times 2 \times 3 \times 3 \times 3 \times 3$

Hence,

$\rm :\longmapsto\: \sqrt{324} = \sqrt{2 \times 2 \times 3 \times 3 \times 3 \times 3}$

$\rm :\longmapsto\: \sqrt{324} = 3 \times 3 \times 2$

$\rm :\longmapsto\: \sqrt{324} = 18$

Additional Information :-

Let's solve same type of problem!!

Question :- Find the smallest whole number by which 7128 should be multiplied to get a perfect square number. Also, find the square root of the number so obtained.

Answer :-

Given number is 7128

Let we first find the prime factorization of 7128

$\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:7128 \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:3564 \:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:1782\:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:891 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:297 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:99 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:33 \:\:}} \\ {\underline{\sf{11}}}& \underline{\sf{\:\:11 \:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}$

So, we have prime factorization of 7128 as

$\rm :\longmapsto\:7128 =2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 2 \times 11$

Now, 7128 have two alone pairs of 2 and 11, so as to make it a perfect square, we have to multiply 7128 by 22.

So required number is 7128 × 22 = 156816

Now,

Prime factorization of 156816 is

$\rm :\longmapsto\:156816 =2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 2 \times 2 \times 11 \times 11$

Hence,

$\rm :\longmapsto\: \sqrt{156816} = \sqrt{2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 2 \times 2 \times 11 \times 11}$

$\rm :\longmapsto\: \sqrt{156816} = 2 \times 2 \times 3 \times 3 \times 11$

$\rm :\longmapsto\: \sqrt{156816} = 396$