Chapter

This method is applicable for squaring of numbers with unit digit 5.  Ekadhikena-purvena literally means by one more than the previous one.

Example 1:  75²   = (7 x 8)/5²

                           =  5625

Step 1:  Multiply the previous number 7 (before 5) by its next number 8 and write the product 56 in the LHS of the answer

Step 2:  Always write 25 in the RHS of the answer

Step 3:  Combine LHS and RHS to obtain the final answer 5625

Note:  There is no carry over in this method.

Example 2:          

15² =  (1 x 2)/5²  

      =  225

25² = 2 x 3/5²

      = 625

35² = (3 x 4)/5²

      =  1225

45² = (4 x 5)/5²

      = 2025

55² =  (5 x 6)/5²

      =  3025

65² = (6 x 7)/5²

      = 4225

75²  = (7 x 8)/5²

       = 5625

85² = (8 x 9)/5²

      =  7225

95²  =  (9 x 10)/25

       =  9025

This Method is applicable for square of any number with unit digit 5

Example  

105² = (10 x 11)/5²

        =  11025

115² =  (11 x 12)/5²

       =   13225

995²  = (99 x 100)/5²

         = 990025

1005²  = (100 x 101)/5²

           = 1010025

Exercises:

125²  = 15625

175²  = 30625

225²  =  50625

37.5²  =  1406.25 (Remember the decimal rule)

245²  =  60025

505²  =   255025

925²   =   855625

1015² =   1030225

1115²  =   1243225

1235²  =   1525225

Corollary:

The principle of Ekadhikena Purvana can also be applied to multiplication of numbers, under certain conditions.

Example 1:  34 x 36 = (3 x 4)/(4 x 6)

                               = 1224 

Here two conditions have to be satisfied.  First, the sum of the unit digits of the numbers should be equal to 10. Secondly, the previous number (before the unit digit) should be same in both the numbers.

Step 1:  Multiply the previous number by 1 more than itself and write the product in LHS of the answer (3 x 4 = 12 in this case)

Step 2:  Multiply the unit digits of both the numbers and write the product in the RHS of the answer (4 x 6 = 24 in this case)

Note that the RHS should always contain two digits.

Step 3:  Combine RHS and LHS to obtain the final answer as 1224 in this case.

Example 2:  22 x 28 = (2 x 3)/(2 x 8)

                               = 616

Example 3:  23 x 27 = (2 x 3)/(3 x 7)

                               = 621

Example 4:  31 x 39 = (3 x 4)/(1 x 9)

                               = 1209 (Note 9 is written is 09 to make it two digits in the RHS)

Example 5:  44 x 46 = (4 x 5)/(4 x 6)

                               = 2024

Example 6:  53 x 57 = (5 x 6) /(3 x 7)

                               = 3021

Example 7:  81 x 89 = (8 x 9)/(1 x 9)

                              = 7209

Example 8:  112 x 118 = (11 x 12)/(2 x 8)

                                  = 13216

Example 9:  244 x 246 = (24 x 25)/(4 x 6)

                                  = 60024

Example 10:  993 x 997 = (99 x 100)/(3 x 7)

                                    = 990021