Who wants to multiply by 9, though? The version of the 13 test I've seen has you multiply by 4 and add instead:
4173 -> 417 + (3 x 4) = 429 -> 42 + (9 x 4) = 78, which is divisible by 13, so 4173 is.
Of course, these two tests work for the same reason. You're saying 10a + b is divisible by 13 iff a - 9b is, I'm saying 10a + b is divisible by 13 iff a + 4b is, and those differ by 13b.
4173 -> 417 + (3 x 4) = 429 -> 42 + (9 x 4) = 78, which is divisible by 13, so 4173 is.
Of course, these two tests work for the same reason. You're saying 10a + b is divisible by 13 iff a - 9b is, I'm saying 10a + b is divisible by 13 iff a + 4b is, and those differ by 13b.