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For denominators like 13, 17 i often see my professor use a method to test whether a given number is divisible or not. The method is not the following : Ex for 17 : subtract 5 times the last digit from the original number, the resultant number should be divisible by 17 etc...

The method is similar to divisibility of 11. He calls it as compartmentalization method. Here it goes.

rule For 17 :

take 8 digits at a time(sun of digits at odd places taken 8 at a time - sum of digits at even places taken 8 at a time)

For Ex : $9876543298765432..... 80$digits - test this is divisible by 17 or not.

There will be equal number of groups (of 8 digits taken at a time) at odd and even places. Therefore the given number is divisible by 17- Explanation.

The number 8 above differs based on the denominator he is considering.

I am not able to understand the method and logic both. Kindly clarify.

Also for other numbers like $13$ and $19$, what is the number of digits i should take at a time? In case my question is not clear, please let me know.

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    I cannot figure out the rule. What are the "blocks of 8 digits at even/odd places"? For example, given the number 1234567890123456, what would the two blocks look like?2012-08-20
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    I think he means the two 8 digit numbers formed by every other digit, in this case $13579135$ and $24680246$.2012-08-20
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    @celtschk: you split into blocks of 8 starting from the ones digit, so the first block would be $90123456$, the second would be $12345678$. Then we have $1234567890123456 \equiv 90123456-12345678 \pmod {17}$2012-08-20

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