Write a proof for the following method for the divisibility

Write a proof for the following method for the divisibility of a number by 17

*****Take any number. Drop the final two digits. Subtract from it nine times the number made by the two digits you dropped. The original number and the new number are either both divisible by 17 or both not divisible by 17

The PROOF should be written in the same format as the example proof provided below.

EXAMPLE only!!! >

The reason this method works is the fact that for any two integers x and y, 19|(10x + y) 19|(x + 2y)

The proof of this is in the following chain of equivalences: 19|(10x + y) 19|2(10x + y) 19|2(10x + y)19x = (x + 2y) The rst step is true because multiplying a number by 2 does not aect the divisibility by 19 (that is, because 2 and 19 are relatively prime). The second step is true because adding a multiple of 19 does not aect divisibility by 19.

Please Help do not just restate what I have already written. To reiterrate the PROOF is to be written for the first method stated for the Divisibility of a number by 17.

Solution

Write your number 10x+y

Then because 10 and 17 are relatively prime,

17x5y1710x50y1710x+y

The last equivalence is because 10x+y(10x50y)=51y is always a multiple of 17.