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Squaring

A square number, also called a perfect square, is an integer that can be written as the square of some other integer. In other words, a number whose square root is an integer is known as the square number of a perfect square.
 
For example, 9 is a square number since it can be written as 3 × 3.
This can be seen through the following flow-chart also.
Description: vm-21.tif 

Properties of a Square Number

  1. The number N is a square number if it can be arranged as N points in a square:
Description: vm-22.tif
 
Thus, it can be deduced that the formula for the nth square number is n2. This is also equal to the sum of the first n odd numbersDescription: 2190.pngas can be seen in the above figure, where a square results from the previous one by adding an odd number of points (marked as ‘’).
 
For example, 52 = 25=1+3+5+7+9.
It should be noted that the square of any number can be represented as the sum 1+1+2+2+…+ n1 + n1+n. For instance, the square of 4 or 42 is equal to 1 + 1 + 2 + 2 + 3 + 3 + 4 = 16. This is the result of adding a column and row of thickness 1 to the square graph of three. This can also be useful for finding the square of a big number quickly. For instance, the square of 52 = 502 + 50 + 51 + 51 + 52 = 2500 + 204 = 2704.
  1. A square number can only end with digits 00, 1, 4, 6, 9, or 25 in base 10, as follows:
  2. If the last digit of a number is 0, its square ends in 00 and the preceding digits must also form a square.
  3. If the last digit of a number is 1 or 9, its square ends in 1 and the number formed by its preceding digits must be divisible by four.
  4. If the last digit of a number is 2 or 8, its square ends in 4 and the preceding digit must be even.
  5. If the last digit of a number is 3 or 7, its square ends in 9 and the number formed by its preceding digits must be divisible by four.
  6. If the last digit of a number is 4 or 6, its square ends in 6 and the preceding digit must be odd.
  7. If the last digit of a number is 5, its square ends in 25 and the preceding digits (other than 25) must be 0, 2, 06, or 56.
  8. A square number cannot be a perfect number. (If the sum of all the factors of a number excluidng the number itself is equal to the number, then the number is known to be a perfect number.)
  9. The digital sum of any perfect square can be only 0, 1, 4, 9, 7. (Digital sum of any number is obtained by adding the digits of the number until we get a single digit. Digital sum of 385 = 3 + 8 + 5 = 1 + 6 = 7)
An easy way to find the squares is to find two numbers which have a mean of it. This can be seen through the following example:
 
To find the square of 21, take 20 and 22, then multiply the two numbers together and add the square of the distance from the mean: 22 × 20 = 440 + 12 = 441. Here, we have used the following formula
(x  y) (x + y) = x2  y2
 
known as the difference of two squares. Thus,
(21–1) (21 + 1) = 212  12 = 440.

Odd and Even Square Numbers

Squares of even numbers are even, since (2n)2 = 4n2.
 
Squares of odd numbers are odd, since (2n + 1)2 = 4(n2 + n) + 1.
 
Hence, we can infer that the square roots of even square numbers are even, and square roots of odd square numbers are odd.





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