Square

In this topic square we are going to see the meaning of squares and example problems.When a number is multiplied by itself twice, the product so obtained is called the square of that number. it is denoted as,"number which is having power 2".

Examples:

12 = 1 x 1 = 1

22 = 2 x 2 = 4

32 = 3 x 3 = 9
If a and b are natural numbers such that a = b2 , then a is a perfect-square of b.

In order to find whether a given number is perfect squre or not, write the number as a product of its prime factors.

1. If a number ends in zero, we can immediately decide whether it is a perfectsqure or not. It may be observed that the number of zeros in the end of a perfect square is never odd.

Examples:

400 is a perfect-square, the number of zeros is two even 4000 is not a perfect square, the number of zeros is three

2. The difference between the squares of two consecutive number is equal to the sum of the numbers

Examples:

25 2 - 24 2 = 25 +24

15 2 - 14 2 = 15 +14

525 2 - 524 2 = 525 + 524

3. If the factors can be grouped in pairs in such a way that both the factors in each pair are equal, we call the given number as a perfect square.

Let us observe the following patterns
22 = 4 = 3 x 1 + 1
32 = 9 = 3 x 3
42 = 16 = 3 x 5 + 1
52 = 25 = 3 x 8 + 1
62 = 36 = 3 x 12
72 = 49 = 3 x 16 + 1
82 = 64 = 3 x 21 + 1
92 = 81 = 3 x 27

22 = 4 = 4 x 1
32 = 9 = 4 x 2 + 1
42 = 16 = 4 x 4
52 = 25 = 4 x 6 + 1
62 = 36 = 4 x 9
72 = 49 = 4 x 12 + 1
82 = 64 = 4 x 16
92 = 81 = 4 x 20 + 1

The above patterns suggest that

(i) The square of a number (other than 1) is either a multiple of 3 or exceeds a multiple of 3 by 1.
(ii) The square of a number (other than 1) is either a multiple of 4 or exceeds a multiple of 4 by 1.

Let us see the example problems below



We observe that the prime factors of 196 can be grouped into pairs as shown and no factors in left over.

196 is a perfect square. It is square of 2 x 7 = 14.






Properties of squares





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