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A perfect square or a square number is a natural number which is the square of another natural number.

Consider the numbers 132 and 225. These can be written as a product of their prime factors as
132 = 2 x 2 x 3 x 11                 225 = 3 x 3 x 5 x 5

The number 132, does not have pairs of identical factors. Thus 132 cannot be a perfect square. 

The number 225, however, has pairs of identical factors.
225 = 3 x 3 x 5 x 5 = 32 x 52 = (3 x 5)2 = 152

Therefore 225 is a perfect square. It is the square of 3 x 5 or 15.
Thus, if all the prime factors of a natural number can be paired, it is a perfect square.

Properties of Squares :-

Property 1 :-  The squares of the first 20 natural numbers are given in the table. Notice that none of them end in 2, 3, 7 or 8. Therefore, we can say that a number ending with 2, 3, 7 or 8 is never a perfect square. Thus 832, 4353, 507 or 10898 are not perfect squares. 


Property 2 :-

There is a simple method to find the unit digit of Square numbers.

Look at the following:


From the above two figures, we can conclude that

If a number ends with 1 or 9, then the unit digit of the square of the number is 1.

If a number ends with 4 or 6, then the unit digit of the square of the number is 6.

Similarly if a number ends with 5, then the unit digit of the square of the number is 5.

Property 3 :-

1) A number ending in an odd number of zeros is never a perfect square. Thus 9530,          42849000, or 85000 are not perfect squares.

2) Note, however, that this does not mean that all numbers ending in an even number of zeros are always perfect squares.

3) They may or may not be perfect squares. Thus 500 is not a perfect square, whereas 400 is a perfect square.

Property 4 :- From the squares of the first twenty numbers, you can notice that
(i) Squares of even numbers are always even.
(ii) Squares of odd numbers are always odd.
This is true for all natural numbers.

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