Properties of Perfect Squares

Friends, we have already posted 3 shortcut techniques for finding Squares. In this post we shall discuss some basic properties of Perfect Squares.

1. A number ending in 2, 3, 7 or 8 is never a perfect square. or all the square numbers end with 0, 1, 4, 5, 6 or 9 at units place. We can verify this statement by observing the squares table here. 
• Note : The converse of the above statement is not true. i.e., if a number ends in 0, 1, 4, 5, 6 or 9, then it is not necessarily a square number. For example, 170, 251, 3584, etc are not Square numbers, though these end with 0, 1, 4.
2. If a number has 1 or 9 in the unit's place, then its square ends in 1. This statement can be verified by observing the squares table.
3. When a square number ends in 6, the number, whose square it is, will have either 4 or 6 in the unit's place. 
4. The number of zeros at the end of a perfect square is always even.
• For Example : 10000 = 100², 2500 = 50², 490000=700²
5. Squares of even numbers are always even and squares of odd numbers are always odd.
• For Example : 2² = 4, 8²=64, 40²=1600, 5²=25,
  9²=81, 17²=289
6. 
   
   For any two consecutive natural numbers n and (n+1), we have

• (n+1) ²-n²   =   (n+1+n)(n+1-n)   =   (n+1)+n
•  For Example :
• 
  
  
  11 ² – 10 ² = 11+10 = 21
  15 ²-14 ² = 15+14 = 29
  19 ²-18 ² = 19+18 = 37 etc
7.  A triplet (x, y, z) of three natural numbers x, y and z is called a Pythagorean triplet if X²  + y²
    = z²    
   
   
• for example, (6, 8, 10) is a Pythagorean triplet.
• Since 6² + 8² =  36+64 = 100
  and 10² = 100
• 
  
  Note : For any natural number n greater than 1, the Pythagorean triplet is given by (2n, n²-1, n²+1)
• 
  
  Example : Find the other two numbers of a Pythagorean triplet, one number of which is 12.
• Solution :
• For any natural number m the Pythagorean Triplet = 2m, m²-1, m²+1 
• let m = 6
• So, 2m = 12
• m²-1 = 6²-1 = 36-1 = 35
• 
  
  m²+1 = 6²+1 = 36+1 = 37
• 
  
  So, the other two numbers of the Pythagorean Triplet are 35 and 37.
8.  The square of a natural number m is equal to the sum of the first m odd numbers. 
• Thus 1² =1 = sum of the first 1 odd number
• 2² = 4 = 1+3 = sum of the first 2 odd numbers
• 
  
  
  3² =9 = 1+3+5 = sum of the first 3 odd numbers
  
  
• 
  
  
  5² =25 = 1+3+5+7+9 = sum of the first 5 odd numbers an so on.
9.  We can express the square of any odd number as the sum of two consecutive positive integers.
• For example, 
• 5² = 25 = 12+13
• 11² = 121 = 60+61
• 41² = 1681 = 840+841 etc
  
  
•  Note : The converse of the above statement is not true. i.e., the sum of any two consecutive positive integers is not necessarily a perfect square or square number.
• Ex : 14+15 = 29, which is not a square number.
10. For any natural number n greater than 1, (n+1) x (n-1) = n²-1
• 
  
  Using this property, we can find the product of two consecutive even or odd natural numbers easily.
• 7 x 9 = (8-1) x (8+1) = 63 =  8²-1
• 15 x 17 = (16-1) x (16+1)  = 255 =16²-1
•  24 x 26 = (25-1) x (25+1) = 624 = 25²-1
11. 
    
    Observe the following :
•  1² = 1 and 2²=4, Between 1² = 1 and 2²=4, the numbers are 2 3.i.e., there are 2 x 1 = 2 non square numbers.
•  2² = 4 and 3² = 9, Between  2² = 4 and 3² = 9, the numbers are 5, 6, 7, 8 i.e., there are 2 x 2 = 4 non square numbers.
• 3² = 9 and 4²  = 16, Between 3² = 9 and 4²  = 16, the numbers are 10, 11, 12, 13, 14, 15. i.e., there are 2 x 3 = 6 non square numbers.
• 8² = 64 and 9²=81, Between 8² = 64 and 9²=81, the numbers are 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80. i.e., there are 2 x 8 = 16 non square numbers.
• Thus, We can say that thre are 2n non perfect square numbers between the squares of the numbers n and (n+1).
12. Study the following pattern :
• 2² = 4 = 3x1+1
• or 2² = 4 = 4x1
  
  
•  3² = 9 = 3x3
• or 3² = 9 = 4x2+1
•  4²= 16 = 3x5+1
• or 4² = 16 = 4x4
• 5² = 25 x 3x8+1
• 5²= 25 = 4x6+1
• 6² = 36 = 3x12
• Or 32 = 36 = 4x9
• 
  
  
  
  From the above we can say that
1. Squares of numbers (greater than 1) can be written as multiples of 3 or multiples of 3 plus 1.
2. Squares of numbers (greater than 1) can also be written as multiples of 4 or multiples of 4 plus 1.

 This property is very useful when we want to check whether a number is a perfect square or not. For example, if we divide a number by 3, and get the remainder 2, then the number is not a perfect square.