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On Divisibility of Numbers

There are some rules to check that a given number is divisible by a number without actual division.
e.g. to check if a given number is divisible by 2, you have just check the last digit is even or not.
Similarly for 3 , if the sum of digit of any given number is divisible by 3 then the number will also be divisible by 3.

One day I was thinking that if there is some generalized rule for every number. Here is some interesting thing I found out.
Suppose we have to test divisibility by 17. Let us check whether 4913 is divisible by 17 or not.

Here is the rule:-
Step 1) :- found out multiple of the number( in this case 17) which ends with 1 or 9.
17 X 3 = 51 ends with 1.
Step 2) :- strip the last digit . so we got 5 here.

Step 3) :- take the last digit of 4913 which is 3. multiply it by 5
3X5=15. Remove the last digit from 4913 and subtract 15 from it.
491-15 = 476
again repeat this on 476
47 - 6X5 = 47 - 30 = 17 which is divisible by 17 . So 4913 is divisible by 17.

Another example : check whether 6859 is divisible by 19 or not.
Step 1) :- found out multiple of the number( in this case 19) which ends with 1 or 9.
19 ends with 9 .
Step 2) :- strip the last digit . so we got 1 here. Add 1 in this we got 2

Step 3) :- take the last digit of 6859 is 9. multiply it by 2
9X2=18. Remove the last digit from 6859 add 18 to it.
685+18 = 703
again repeat this on 703
70 + 3X2 = 70 + 6 = 76 which is divisible by 19 . So 6859 divisible by 17.

That's it folks ,bye

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