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Showing posts with label Aptitude Shortcuts. Show all posts
Showing posts with label Aptitude Shortcuts. Show all posts

Monday, 18 November 2013

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.
Read shortcut techniques for finding squares from here
21:49 - By Unknown 0

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