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 nal 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.

Let see how this works for the rst example above, the number 20,986,619.
209866 -9 *19 = 209695
2096 -9 *95 = 1241
12 -9 *41 = -357
3 -9 * 57 = -510
5 -9 * 10 = -85
Since the last number is divisible by 17 the rst one also is.

Solution

Solution: 03/07/2016.

Let see how this works for the rst example above, the number 20,986,619.
209866 -9 *19 = 209695
2096 -9 *95 = 1241
12 -9 *41 = -357
3 -9 * 57 = -510
5 -9 * 10 = -85
Since the last number is divisible by 17 the rst one also is.

= 8 - 9 x 5 = 8 - 45

= -37

Since the last number is not divisible by 17.