Perfect Square


 
 
Concept Explanation
 

Perfect Square

Square of a number and square numbers:

The square of a number is that number raised to the power 2.

Thus, if a is a number, then the square of a is written as a^{2} and is given by

                a^{2}=atimes;a

That is, the square of a number is obtained by multiplying it by itself

If atimes;a=b  i.e. a^{2}=b, then we say that the square of number a is number b or the number b is the square of number a.

For example,

      2^{2}=2times2=4, so we say that the square of 2 is 4

     3^{2}=3times3=9, so we say that the square of 3 is 9

     12^{2}=12times12=144, so we say that the square of 12 is 144 etc.

Perfect Square or Square Number:

A natural number n is called a perfect square or a square number if there exist a natural number m such that n=m^{2}.

In other words, a natural number is a perfect square if it is the square of the same natural number.

Since 4=2times2=2^{2}. Therefore, 4 is a square number.

Also,

    25=5times5=5^{2}, so 25 is a square number or a perfect square.

   144=12times12=12^{2}, so 144 is a square number or a perfect square etc.

STEP I  Obtain the natural number.

STEP-II  Write the number as a product of prime factors.

STEP III Group the factors in pairs in such a way that both the factors in each pair are equal.

STEP IV See whether some factor is leftover or not. If no factor is left over in grouping, then the given number is a perfect square. Otherwise, it is not a perfect square.

STEP V  To obtain the number whose square is the given number take one factor from each group and multiply them.

Illustration: Show that 17640 is not a perfect square.

Solution: Resolving 17640 into prime factors, we have

                

                 17640=2times2times 2times 3times 3times 5times 7times 7

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Sample Questions
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Question : 1

The smallest number by which 136 must be multiplied so that it becomes a perfect square is __________________.

Right Option : C
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Question : 2

How many numbers lie between square of 18 and 19?

Right Option : C
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Question : 3

A teacher arranges his students in such a way that each row contains as many students as there are rows. If in each row there are 80 students, and 20 students remained in the end what was the total number of students ?

Right Option : D
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