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 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)

DIVISIBILITY BY 13

Take any number. Drop the nal 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.
Let see how this works for the rst example above, the number 11,264,331.

112643 + 3 31 = 112736
1127 + 3 36 = 1235
12 + 3 35 = 117
1 + 3 17 = 52
Since the last number is divisible by 13 the rst one also is.

Solution

. Test for divisibility by 13.

Add four times the last digit to the remaining leading truncated number. If the result is divisible by 13, then so was the first number. Apply this rule over and over again as necessary.
Example:

50661==>5066+4=5070==>507+0=507==>50+28=78 and 78 is 6*13, so 50661 is divisible by 13.