Suppose we have a no P = 1234567. Now if we divide this no in X and Y such that X = 123456 and Y = 7. So all the following things are based on that assumption that P is divided in XY where Y is units digit and X is rest of the no.

We see the divisiblity test of 19 and 29 i.e.: For 19 Divisibility Test is X+2Y. i.e. if X+2Y is divisible by 19 then P is divisible by 19. And for 29 it is X+3Y. i.e. if X+3Y is divisible by 29 then P is divisible by 29. We can see a pattern here and come up with this:

I write the following no on a piece of paper and their Divisibility below them following a particular pattern...

M: 9 19 29 39 49 59 69 79 89

N: X+Y X+2Y X+3Y X+4Y X+5Y X+6Y X+7Y X+8Y X+9Y

ie. a no P is divisible by M if N is divisible by M.

I takek various examples at random and found out that the above statement is true....

Done that I then thought of doing something similar for numbers ending with 1, 3 and 7. So within half an hour I came up with the following divisibility tests....

M: 3 13 23 33 43 53 63 73 83 93

N: X+Y X+4Y X+7Y X+10Y X+13Y X+16Y X+19Y X+22Y X+25Y X+28Y

M: 7 17 27 37 47 57 67 77 87 97

N: X-2Y X-5Y X-8Y X-11Y X-14Y X-17Y X-20Y X-23Y X-26Y X-29Y

M: 11 21 31 41 51 61 71 81 91

N: X-Y X-2Y X-3Y X-4Y X-5Y X-6Y X-7Y X-8Y X-9Y

If we observe all the Divisibility series follow an AP series.

Then I thought if this is true if the no whose divisibility test we are finding is greater than 100. e.g. we have to find if a no P is divided by 131 or 149 something like that. And I found out that if we follow this method it is indeed true even for nos greater than 100.

e.g = Lets say 131 * 23 = 3013. Therefore 131 must be a factor of 3013. Lets see...

7647: X = 301 Y = 3

Following the series Divisibility Test for 131 will be X - 13Y

Therefore 301 - (13 * 3) = 301 - 39 = 262 which is divisible by 131.

Again for 149 : 149 * 37 = 5513

X = 551 and Y = 3.

Following the series Divisibility Test for 149 will be X + 15Y

Therefore 551 + (15 * 3) = 551 + 45 = 596 which is divisible by 149.

Hence in General I found out that to find the divisibility of any number whose end values are 1,3,7 or 9 we have the following test:

If Number is A1 where 1 is unit place and A is Ten's place Divisibility Test will be: X - (A * Y)

If Number is A3 where 3 is unit place and A is Ten's place Divisibility Test will be: X + [(1+2A)Y]

If Number is A7 where 7 is unit place and A is Ten's place Divisibility Test will be: X - [(3A-1)Y]

If Number is A9 where 9 is unit place and A is Ten's place Divisibility Test will be: X + (A * Y)

Interesting isnt it??