Fun with Mathematics

Fun with Mathematics-2

The product with all identical digits

According to the divisibility rule, a number N is divisible by 3,

if the sum of all the digits of N is divisible by 3. Hence the

number with treble, six, nine or twelve 1’s will be divisible by 3.

Making use of this fact, one can create products with all identical

digit 1.

                                    37 x 3 = 111

                            37 0 37 x 3 = 111 111

                    37 0 37 0 37 x 3 = 111 111 111

            37 0 37 0 37 0 37 x 3 = 111 111 111 111

Since 111 x n = nnn ,we have 37 x 3 x n = nnn, where n is a

single digit number. This n may be combined with the multiplier

or with the multiplicand and as a consequence of which, the

product with all identical digit can be represented in different ways.

          

           37 x 6 = 222          74 x 3 = 222

           37 x 9 = 333        111 x 3 = 333

           37 x 12 = 444      148 x 3 = 444

           37 x 15 = 555      185 x 3 = 555

           37 x 18 = 666      222 x 3 = 666

           37 x 21 = 777      259 x 3 = 777

           37 x 24 = 888      296 x 3 = 888

           37 x 27 = 999      333 x 3 = 999

For numbers having 4,6,8,…….2n (where n ≥ 2) identical digits,

a number with the same two identical digits will be a common

factor. That is 11 is a common factor for 1111,111111 and so on.

                                 1 x 11 = 11

                           1 0 1 x 11 = 11 11

                         10101 x 11 = 11 11 11

                     1010101 x 11 = 11 11 11 11

For every annexes of 10 in the left or 01 in the right of the

multiplicand, one block number of 11 is added in the product.

To produce products with any required identical digits, we have

                          101 x 11 x n = nnnn

As n may be combined either with the multiplicand or with the

multiplier

                        101 x nn = n0n x 11 = nnnn

 Similarly for the numbers having 6,9,…. 3n (where n ≥ 2)

identical digits, a number with the same three identical digits

will be a common factor. That is 111 is a common factor for 111,111; 111,111,111;   and so on.

                       1001 x 111 = 111 111

                100 1001 x 111 = 111 111 111

         100 100 1001 x 111 = 111 111 111 111

Following a similar procedure, it can be shown as,

                          1 x 1111 = 1111

                 1000 1 x 1111 = 1111 1111

        1000 1000 1 x 1111 = 1111 1111 1111

 and

                          1 x 11111 = 11111

               10000 1 x 11111 = 11111 11111

    10000 10000 1 x 11111 = 11111 11111 11111

41 and 271 are the two prime factors to 11111

                        41 x 271 = 11111

By suitably introducing intermediate zeros either with the

multiplicand or with the multiplier, the block number 11111

can be increased.

                       41 000 41 x 271 = 11111 11111

                       41 x 271 00 271 = 11111 11111

111 111 has many factors and hence it can be expressed

as a product of two numbers by many ways.

15873 x 7       ]

10101 x 11     ]

8547   x 13     ]

5291   x 21     ]

3367   x 33     ]

3003   x 37     ]

2849   x 39     ]       = 111 111

1443   x 77     ]

1221   x 91     ]

1001   x 111   ]

777     x 143   ]

429     x 259   ]

273     x 407   ]

231     x 481   ]

 The number with seven successive 1’s can be expressed

  239 x 4649 = 1 111 111

The number 1111 1111 has many factors and hence it

can be expressed by many ways as the product of its two factors.

1010101 x 11      ]

152207   x 73      ]

110011   x 101    ]

81103     x 137    ] = 1111 1111

13837     x 803    ]

7373       x 1507  ]

1111       x 10001]

 In the case of nine successive 1’s,

37037037  x 3     ]

12345679  x 9     ]

3003003    x 37   ]  = 111 111 111

1001001    x 111 ]

333667      x 333 ]