How do you prove that the set of decimal representation of the 4 divisble natural numbers is regular?

Question

This is from an old exam, the last Task no one could solve correctly and I'm curious how it's done :p

Show that the set of decimal representation (without leading zeroes) of the divisible numbers by 4 (natural numbers) is regular.

By this thread How to prove a language is regular? I know that one can make a DFA to Show that a language is regular.

But is that possible at all because we have infinite natural numbers that are divisible by 4..

I cannot even imagine how that DFA would look like :o

Maybe there is another way of showing this, too?

Edit: Removed..

First answer how would a number divisible by 4 look like in binary. Hint: assuming that first input bit is LSB, the minimal DFA has only 3 states. — rus9384, Oct 19 '17 at 08:51
@rus9384 I think I understand your hint but will Need 4 states. Could you check it if I edit my question in about 20 minutes? — cnmesr, Oct 19 '17 at 09:03
Well, if there are 4 states, it's not minimal (even if first bit is MSB). Accepting state can have transitions to non-accepting states. — rus9384, Oct 19 '17 at 09:07
This is a special case of https://cs.stackexchange.com/questions/640/language-of-the-values-of-an-affine-function — Gilles 'SO- stop being evil', Oct 19 '17 at 12:53
@cnmesr We don't check people's work. That's what your TA is for. — David Richerby, Oct 20 '17 at 00:07

score 4 · Accepted Answer · edited Oct 19 '17 at 12:54

4

A decimal number is divisible by 4 if the number formed by its last two digits are divisible by 4. Stated differently, divisibility by 4 depends only on the last two digits. That should be enough for you to show that your language is regular.

The language of numbers in base $b$ divisible by $m$ is regular for all $b,m$, but that's somewhat harder to show.

edited Oct 19 '17 at 12:54

Gilles 'SO- stop being evil'

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answered Oct 19 '17 at 11:42

Yuval Filmus

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Can you please check my Edit? I've used your explanation and the given link and had created it like that. But I don't know how to continue :s I hope the part I created is correct at least? – cnmesr Oct 19 '17 at 23:42
Unfortunately I cannot replace your TA. – Yuval Filmus Oct 20 '17 at 00:07

How do you prove that the set of decimal representation of the 4 divisble natural numbers is regular?

1 Answers1