Fun with mathematics

Fun with Mathematics-1

Multiplicand with identical digits and its product with a multiplier.

At first let us consider a simple example with a single digit multiplier.

             Multiplicand x multiplier = product

               2222… (m) x 4               = 8  8888888… (m-2)  8

               2222… (m) x 7               = 1  5555555… (m-1)  4

               2222… (m) x 9               = 1  9999999… (m-1)  8

It is noted that the two edge numbers are derived from the product of the multiplier (n) and the identical digit of the multiplicand, in our case it is 2.  If the supplemental product has only two digits, its right digit is placed in the right end and its left digit is placed in the

left end of the prime product. If the multiplicand has m identical digits, the sum of the digits of the supplemental product is placed

(m-1) times in between them.

If the supplemental product has more than two digits, then its right

digit is placed as before in the right end while the remaining block number  having two or more  digits is placed in the left end. The sum of these two edge numbers is determined. Its right digit is placed in right adjacent to the right end, while the remaining left portion is added with the block number in the left end. It is repeated till the subsequent sum results with a single digit. If the multiplicand has m identical digits and the right end block number

has x digits, then there will be (m-x) identical digits in the middle .

It is illustrated below.

                 2222… (m) x 14 =  2……………8

Since 2 x 14 = 28, the left and right end numbers of the prime product become 2 and 8 respectively. Its sum is 2 + 8 = 10, its left digit is added with the left end block number, while the right digit

is placed adjacent to the right end number. Hence the new edge numbers become 3 ……… 08. Since the sum gives a two digit number, subsequent summation is continued. 1 + 0 = 1. It exists

(m-2) times in the middle between the end numbers.

                 2222… (m) x 14 = 3  11111… (m-2) 08

                 2222… (m) x 19 = 4  22222… (m-2) 18

                 2222… (m) x 23 = 5  11111… (m-2) 06

                 2222… (m) x 24 = 5  33333… (m-2) 28

                 2222… (m) x 52 =11 55555… (m-2) 44

It provides an unfamiliar method of multiplication associated with all identical digit multiplicand. The multiplication becomes easy for bigger multiplier. Two specific examples are given below.

3333 …. (m) x 1234 = ?

The supplemental product … 3 x 1234 = 3702;     370 ………. 2

Subsequent summation ….     370 + 2 = 372   ;     407……….22

                                                37 + 2 = 39       ;     410………922

                                                3+9 = 12           ;     411……..2922

                                                1+2 = 3  ;  411 3333… (m-4) 2922

                 3333… (m) x 1234 = 411  3333… (m-4) 2922

7777… (m) x 3462  = ?

The supplemental product  7 x 3462 = 24234 ;     2423 ……… 4

Subsequent summation   2423 + 4 = 2427  ;             2665 ….     74

                                           242 + 7 = 249   ;            2689 …..   974

                                            24+9 = 33        ;            2692 ….. 3974

                                            3 + 3 = 6  ; 2692  66666 …(m-4) 3974

         7777 … (m) x 3462 = 2692  666666… (m-4) 3974