Write a proof for the following method for the divisibility

******Write a proof for the following method for the divisibility of a number by 13, the format should be written in the same style as the example shown at the bottom.

******Take any number. Drop the final two digits. Add to it three times the number made by the two digits you dropped. The original number and the new number are either both divisible by 13 or both not divisible by 13.

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)

Please Help, do not just restate what I have written above. The PROOF is to be written for the method described for the divisibility of a number by 13.

Solution

for suppose i have taken the number as 169.

by following the above conditions;

here 100 is after dropping the last two digit no.\'s from 169

and 207 is three times the dropped no.(i.e. 69)

100+ 207 = 307

169/13 = 13 ,so 169 is divisible by 13

307/13= 23.615 , so 307 is not divisible by 13