Question

•By which least number should 2352 be divided so that it is a perfect square? Find the perfect square number. Also, find the square root.

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Answer 1

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

Given number 2352

Since, we have to find the least number by which 2352

must be divided to make it a perfect square.

So, we use prime factorization method to find

the number by which 2352 should be divided  to make it a perfect square.

Thus,

Prime factorization of 2352

$\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:2352 \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:1176 \:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:588\:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:294 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:147 \:\:}} \\ {\underline{\sf{7}}}& \underline{\sf{\:\:49 \:\:}} \\ {\underline{\sf{7}}}& \underline{\sf{\:\:7 \:\:}} \\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}$

So, Prime factorization of 2352 is

$\red{\rm :\longmapsto\:\sf{ \: 2352 = 2 \times 2 \times 2 \times 2 \times 3 \times 7 \times 7}}$

So, to make 2352 to be perfect square, we find that 3 is the only lone pair number.

So, 2352 must be divided by 3, to make it a perfect square.

So, Required number is 2352 ÷ 3 = 784

So, Prime factorization of 784 is

$\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:784 \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:392 \:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:196\:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:98 \:\:}} \\ {\underline{\sf{7}}}& \underline{\sf{\:\:49 \:\:}} \\ {\underline{\sf{7}}}& \underline{\sf{\:\:7 \:\:}} \\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}$

So, Prime factorization of 784 is

$\red{\rm :\longmapsto\:\sf{ \:784 = 2 \times 2 \times 2 \times 2 \times 7 \times 7}}$

$\rm :\longmapsto\: \sqrt{784} = \sqrt{2 \times 2 \times 2 \times 2 \times 7 \times 7}$

$\rm :\longmapsto\: \sqrt{784} = 2 \times 2 \times 7$

$\rm :\longmapsto\: \sqrt{784} = 28$

Answer 2

Answer:

28

Step-by-step explanation:

2352=2^4*7^2*3

So it must be divided by 3

So, required number = 784

sqrt(784) = 28