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..

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    $\begingroup$ 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. $\endgroup$ – rus9384 Oct 19 '17 at 8:51
  • $\begingroup$ @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? $\endgroup$ – cnmesr Oct 19 '17 at 9:03
  • $\begingroup$ 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. $\endgroup$ – rus9384 Oct 19 '17 at 9:07
  • $\begingroup$ This is a special case of cs.stackexchange.com/questions/640/… $\endgroup$ – Gilles 'SO- stop being evil' Oct 19 '17 at 12:53
  • $\begingroup$ @Gilles Please check my edit $\endgroup$ – cnmesr Oct 19 '17 at 23:43

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.

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  • $\begingroup$ 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? $\endgroup$ – cnmesr Oct 19 '17 at 23:42
  • $\begingroup$ Unfortunately I cannot replace your TA. $\endgroup$ – Yuval Filmus Oct 20 '17 at 0:07

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