Square Numbers - III

Some interesting facts

• There are 2n non-perfect square numbers between the squares of two consecutive natural numbers n and (n+1) e.g. between 2² and 3² , there are  

         2 x 2 = 4 , non-perfect square numbers.

•  The square number of an odd natural number n , can be expressed as the sum of two consecutive natural numbers (n²-1)/2 and (n²+1)/2 e.g.  

          5² = (5²-1)/2 + (5²+1)/2.

• A triplet of three natural numbers a , b and c forms a Pythagorean Triplet , if a²+b²=c² e.g. (3,4,5) is a Pythagorean Triplet. For any natural number p greater than 1 , (2p , p²-1 , p²+1) is a Pythagorean triplet. 
• The squares of numbers which have all the digits as 1 , exhibit the following pattern

 

• Addition of two consecutive triangular numbers exhibits the following pattern :

             1 + 3 = 4 = 2²
             3 + 6 = 9 = 3²
             6 + 10 = 16 = 4²
This shows that if we add two consecutive triangular numbers we get a square number.  

Square Numbers - II

Properties of Square Numbers

• A number that ends in 2 , 3 , 7 or 8 is never a perfect square. 
• The ones digit in the square of a number can be determined if the ones digit of the number is known. Look at the following table

                                 

• The number of zeroes at the end of a perfect square is always even e.g.                40² = 1600 , 200² = 40000
• The square of an even number is always an even number and square of an odd number is always an odd number. e.g. 12² = 144 , 23² = 529
• If n is a perfect square then 2n can never be a perfect square. e.g. 100 is a perfect square , but 2 x 100 = 200 is not a perfect square number. 
• The difference between the squares of the two consecutive numbers is equal to their sum or twice the smaller number plus 1. e.g. 

                     (m+1) ² - m² = [(m+1) + m] [(m+1) – m] = (2m +1)

• If a number is a square number , it has to be the sum of successive odd numbers starting from 1. e.g. 

               1 + 3 = 4 = 2²                          [Sum of first two odd numbers]   
               1 + 3 + 5 + 7 + 9 = 25 = 5²       [ Sum of first five odd numbers]

• The square of a number , either negative or positive is always positive.          e.g. (-3) ² = (-3) x (-3) = 9 
• The square of a natural number other than 1 is either a multiple of 3 or exceed the multiple of 3 by 1. Thus we can express square of a number (other than 1) as 3m or 3m+1 for some natural number m. e.g.

                      5² = 25 = 3 x 8 + 1 
                    12² = 144 = 2 x 36 

• The square of a natural number other than 1 is either a multiple of 4 or exceed the multiple of 4 by 1. Thus we can express square of a number (other than 1) as 4m or 4m+1 for some natural number m. e.g.

                    7² = 49 = 4 x 12 + 1 

                 14² = 196 = 4 x 49