Show that all numbers of the form 12008, 120308, 1203308, 12033308, … are divisible by 19

Let the nth number be

100x + 8

the (n+1)st number be 1000x + 308

the difference is 900x + 300 = 300(3x + 1)

if 100x + 8 = 0 mod 19

then 5x + 8 = 0 mod 19
20x + 32 = 0 mod 19
or x + 13 = 0 mod 19
or 3x + 39 = 0 mod 19
or 3x +1 = 0 mod 19
so difference is divisible by 19 if nth number is divisible by 19

1st number is divisible by 19 and hence the difference and and so the second one

using the logic above we see that next one and so so
I have proved by mathematical induction