Introduction

☼ Square of  a number : If a number is multiplied by itself the product so obtained is called the square of  that number.

 e.g. Square of  9  =  9  9  =  81

■ Perfect Square : The square of  a natural number is called a perfect square.

■ Some Properties of  Square Numbers :

 ▪ A square  cannot  end  with  an  odd  number  of  zeros.

 ▪ A square number cannot end with  2, 3, 7 or  8.

 ▪ If  the given number has  n  digits, the square must contain   2n  or   2n – 1  digits.

 ▪ In cases of  pure decimals, the number of  digits in the square is always double that in the square root.  ( e.g.  =  0.01,    =  0.0625 )

 ▪ The square of  an odd number is odd.

 ▪ The square of  an even number is even.

 ▪ Every square number is a multiple of  3, or exceeds a multiple of  3  by unity.

i.e.   every square number  =  3m   or   3m + 1

 ▪ Every square number is a multiple of  4, or exceeds a multiple of  4  by unity.

i.e.   every square number  =  4m   or   4m + 1

 ▪ 1, 5, 6 and 0 at the end of  a number reproduce themselves as the last digits in their squares.

 ▪ If a square number ends in 9, the preceding digit is even.

 ▪ If a square number ends in 6, the preceding digit is odd.

■ Square of a Number Ending in 5:

Multiply  the number of  tens by the next higher integer and annex 25 to the right of  the product .  

 e.g. 125² :  12  13 = 156.  Hence square is 15625.

■ Square of  a Number Ending in 25:

Multiplying the hundred’s digit by a number consisting  of  the hundred’s digit with a 5 to its right and annex 625 to the product.

 e.g. 425²  :   4  45 = 180.  Hence square is 180625.

325²  :   4  35 = 105.  Hence square is 105625.

■ Square of   1 ,  2 ,  3  etc:

 Multiplying the integral portion by the next higher integer and add

 e.g.        Or   

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