Question

Find the smallest number by which 500 must be multiplied so that it becom
perfect square. Also, find the square root of the number so obtained.
​

Answer 1

Answer :-

• The smallest number by which 500 must be multiplied so that it becomes a perfect square is 5.
• The square root of the number so obtained is 50.

Solution :-

• Let's find the smallest number by which 500 should be multiplied so that it becomes a perfect square by prime factorisation!

Prime factorisation of 500 :-

$\begin{array}{c | c} \underline5 & \underline{500} \\ \underline5& \underline{100} \\ \underline5& \underline{ \: 20} \\ \underline{2}& \underline{ \: \: 4} \\ \underline2& \underline{ \: \: 2} \\ & \: \: 1\end{array}$

The given number is 500.

It can be expressed as :-

$\sf500 = \underbrace{5 \times 5} \times 5 \times \underbrace{2 \times 2}$

Since :-

• 5 is left unpaired, hence 5 is the smallest number by which 500 must be multiplied so that it becomes a perfect square.

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Now,

• 500 × 5 = 2500.

So,

• We have to find the square root of 2500.

Prime factorisation of 2500 :-

$\begin{array}{c | c} \underline2 & \underline{2500} \\\underline2&\underline{1250} \\ \underline5 & \underline{625} \\ \underline5& \underline{125} \\\underline5&\underline {25}\\\underline5&\underline5 \\ &1 \end{array}$

Now,

$2500 = \underbrace{2 \times 2} \times \underbrace{5 \times 5} \times \underbrace{5 \times 5}$

$2500 = {2}^{2} \times {5}^{2} \times {5}^{2}$

$2500 = (2 \times 5 \times 5)^{2}$

$100 = (50) ^{2}$

Hence :-

• The square root of 2500 is 50.

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Answer 2

Question:

• Find the smallest number by which 500 must be multiplied so that it becomes perfect square.Also,find the square root of the number so obtained.

Solution:

Taking LCM of 500

2 | 500

2 | 250

5 | 125

5 | 25

5 | 5

1

• LCM of 500 = 2 × 2 × 5 × 5 × 5

Here,

• The required number is 5.

Therefore,

• 500 = 2 × 2 × 5 × 5 × 5 × 5

By multiplying we get,

• 500 = 2 × 2 × 5 × 5 × 5 × 5 = 2500

Let us square root 2500,we get:

• √2500 = 50

Thus,

• 2500 is a perfect square.

Answer:

• 2500 is a perfect square and multiplied number is 5.