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

__________________________________________________________

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