creative thoughts-5

Creative thoughts-5 (26-th July 2010)

1.A thought to think

Things which are used excessively will lose its credits,if
they are used less,they will be useless after sometime.
This is true for everything even to our knowledge.

2.Recreational mathematics
Divisibility test

Here is an unique method to test the divisibility of any
multi-digit numbers by any multi-digit divisors.

A divisor (D) has one characteristic feature called smallest
index multiplier I. Its higher index (I') multipliers are the sum of the
smallest index multiplier and any multiples of the divisor
                         I' = I + nD
It is simple to find out I for any D.For example let the
divisor be 6.Few multiples of 6 are 12,18,24,30,36......Now
find out which number when multiplied with the block number
of them excluding the digit of units and added with it gives
again a multiple of the divisor.
1x4+2 =6; lx4+8=12; 2x4+4=12 ;3x4+0 =12; 3x4+6=18
That is for the divisor 6, 4 is its smallest index multiplier and its
higher index multipliers are 10,16,22.... which are simply denoted
by (4+6n). It is noted that the index multiplier for the divisor 6 is
independent of the way by which the given number under test
is separated into right and left blocks.e.g.,117876

d=l;       6 + 4x11787  = 47154 = 6 x 7859
d=2;    76 + 4x1178    =   4788 = 6 x 798
d=3;  876 + 4x117      =   1344 = 6 x 224
d=4;7876 + 4x11        =   7920 = 6 x 1320

For some other divisors, the index multiplier will be different
according to the number of digits in the left block number
after separation.

This method can be used for testing the divisibility of the given
number however big it is. By repeatedly doing this process,
the given number can be reduced smaller convenient for testing.
In the above typical example 117876,
      step.1:   6 + 4 x 11787 = 47154 = 6 x 7859
      step.2:   4 + 4 x 4715   = 18864 = 6 x 3144
      step.3:   4 + 4 x 1886   =   7548 = 6 x 1258
      step.4:   8 + 4 x   754   =   3024 = 6 x   504
      step.5:   4 + 4 x   302   =   1212 = 6 x   202
      step.6:   2 + 4 x   121   =     486 = 6 x     81
      step.7:   6 + 4 x     48   =     198 = 6 x     33
      step.8:   8 x 4 x     19   =       84 = 6 x     14
However this process is laborious. To reduce a bigger number
under testing into a conveniently smaller number one can subtract
any multiples of the divisor either from the right or from the left
or from the both block numbers.It is exemplified with the same
example. Instead of taking 6 + 4 x 11787, the bigger multiplicand
is conveniently reduced by  subtracting 6x. To predict a bigger
number which is divisible by 6 and  closest to the multiplicand
it must be even and its digital root must be three.This is achieved
by attaching 4 with 1178, that is 11784 is closest to 11787 and is a
multiple of 6 as well. Hence
6 + 4 (11787 - 11784) = 6 + 4 x 3 = 18 = 6 x 3
Similarly instead of taking 7876 + 4 x 11, we can consider
(7876-7872) + 4 x 11 = 4 + 44 = 48 = 6 x 8
Reducting of digit can also be used for this purpose. Any digit
greater than 6 in the bigger multiplicand can be reduced by 6. It helps
to reduce the multiplicand conveniently to a smaller number within no
time.In the same typical example,
6 + 4 x 11787 --> 6 + 4 x 5121 (using 11-6=5,7-6=1;8-6=2)
6 + 4 x 5121 --> 6 + 4 x 303 (using 21-18=03,51-48=3)
6 + 4 x  303 --> 6 + 4 x 3 (using 30-30 =0) 18 =6 x 3
The table given below gives the values of lowest positive index
multiplier for the summation with the last two digit block number
of the given number under divisibility test.
_________________________________________________________
I st digit/last digit  1       2       3       4       5       6       7       8       9
_________________________________________________________
      0                     ...       0       1       2       5       4       3       2       1
      1                      1       4       9       2     10       4     15     10       5
      2                    16     12       8       4     25     22     19     16     13
      3                      7       4       1     32     30     28     26     24     22
      4                    18     16     14     12     10       8       6       4       2
      5                    49     48     47     46    45      44     43     42     41
      6                    39     38     37     36    35      34     33     32     31
      7                    29     28     27     26    25      24     23     22     21
      8                    19     18     17     16    15      14     13     12     11
      9                      9       8       7       6      5        4       3       2       1
__________________________________________________________