> The result will be a smaller number which has the same mod-19 residue.
It seems to me that the procedure works only for numbers divisible by 19. If it's not divisible, the residue may change. Try 20 for example.
It seems to me that the procedure works only for numbers divisible by 19. If it's not divisible, the residue may change. Try 20 for example.
Take a number say 123. Write it as 1 * 10^2 + 2 * 10 + 3.
In the first step we multiply parts of this expression by 20. This doesn't change the residue since 20 = 1 mod 19
1 * 10^2 + 2 * 10 + 3
1 * 10^2 + (2+3*2) * 10
1 * 10^2 + 8 * 10
(1+8*2) * 10^2
17 * 10^2
20 * 2 === 40 === 2 (mod 19)
21 * 2 === 42 === 4 (mod 19)
18 * 2 === 36 === 17 (mod 19), equivalently -1 * 2 === -2 === 17 (mod 19)
10 * 2 === 20 === 1 (mod 19), equivalently -9 * 2 === -18 === 1 (mod 19)