Check number divisibility with regular expressions

Question

Given a decimal number N as a string of digits, how do I check if it\'s divisible by M using regular expressions only, without con

Answer 1

This is an old question, but none of the existing answers include code. I wrote code to compute these regular expressions back in 2010, which is online here and has a link to the commented source code, so I thought it might be useful to add a link to it here.

The technique is basically the one described in eh9’s answer, except that I compute a regular expression directly from the DFA using state elimination and then apply some simplifications to the result. They’re plain regular expressions, without recursion. (Using recursion would make the results much shorter, but I think it’s interesting that it’s possible without.)

The results aren’t quite as verbose as eh9’s estimate would suggest. For example, here is a regular expression that matches multiples of 7 in base 10:

^(0|7|[18]5*4|(2|9|[18]5*6)(3|[29]5*6)*(1|8|[29]5*4)|(3|[18]5*[07]|(2|9|[18]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(5|65*4|(0|7|65*6)(3|[29]5*6)*(1|8|[29]5*4))|(4|[18]5*[18]|(2|9|[18]5*6)(3|[29]5*6)*(5|[29]5*[18])|(3|[18]5*[07]|(2|9|[18]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))(6|35*[18]|(4|35*6)(3|[29]5*6)*(5|[29]5*[18])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))*(2|9|35*4|(4|35*6)(3|[29]5*6)*(1|8|[29]5*4)|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(5|65*4|(0|7|65*6)(3|[29]5*6)*(1|8|[29]5*4)))|(5|[18]5*[29]|(2|9|[18]5*6)(3|[29]5*6)*(6|[29]5*[29])|(3|[18]5*[07]|(2|9|[18]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(3|65*[29]|(0|7|65*6)(3|[29]5*6)*(6|[29]5*[29]))|(4|[18]5*[18]|(2|9|[18]5*6)(3|[29]5*6)*(5|[29]5*[18])|(3|[18]5*[07]|(2|9|[18]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))(6|35*[18]|(4|35*6)(3|[29]5*6)*(5|[29]5*[18])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))*(0|7|35*[29]|(4|35*6)(3|[29]5*6)*(6|[29]5*[29])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(3|65*[29]|(0|7|65*6)(3|[29]5*6)*(6|[29]5*[29]))))(4|[07]5*[29]|(1|8|[07]5*6)(3|[29]5*6)*(6|[29]5*[29])|(2|9|[07]5*[07]|(1|8|[07]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(3|65*[29]|(0|7|65*6)(3|[29]5*6)*(6|[29]5*[29]))|(3|[07]5*[18]|(1|8|[07]5*6)(3|[29]5*6)*(5|[29]5*[18])|(2|9|[07]5*[07]|(1|8|[07]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))(6|35*[18]|(4|35*6)(3|[29]5*6)*(5|[29]5*[18])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))*(0|7|35*[29]|(4|35*6)(3|[29]5*6)*(6|[29]5*[29])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(3|65*[29]|(0|7|65*6)(3|[29]5*6)*(6|[29]5*[29]))))*(6|[07]5*4|(1|8|[07]5*6)(3|[29]5*6)*(1|8|[29]5*4)|(2|9|[07]5*[07]|(1|8|[07]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(5|65*4|(0|7|65*6)(3|[29]5*6)*(1|8|[29]5*4))|(3|[07]5*[18]|(1|8|[07]5*6)(3|[29]5*6)*(5|[29]5*[18])|(2|9|[07]5*[07]|(1|8|[07]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))(6|35*[18]|(4|35*6)(3|[29]5*6)*(5|[29]5*[18])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))*(2|9|35*4|(4|35*6)(3|[29]5*6)*(1|8|[29]5*4)|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(5|65*4|(0|7|65*6)(3|[29]5*6)*(1|8|[29]5*4))))|(6|[18]5*3|(2|9|[18]5*6)(3|[29]5*6)*(0|7|[29]5*3)|(3|[18]5*[07]|(2|9|[18]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(4|65*3|(0|7|65*6)(3|[29]5*6)*(0|7|[29]5*3))|(4|[18]5*[18]|(2|9|[18]5*6)(3|[29]5*6)*(5|[29]5*[18])|(3|[18]5*[07]|(2|9|[18]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))(6|35*[18]|(4|35*6)(3|[29]5*6)*(5|[29]5*[18])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))*(1|8|35*3|(4|35*6)(3|[29]5*6)*(0|7|[29]5*3)|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(4|65*3|(0|7|65*6)(3|[29]5*6)*(0|7|[29]5*3)))|(5|[18]5*[29]|(2|9|[18]5*6)(3|[29]5*6)*(6|[29]5*[29])|(3|[18]5*[07]|(2|9|[18]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(3|65*[29]|(0|7|65*6)(3|[29]5*6)*(6|[29]5*[29]))|(4|[18]5*[18]|(2|9|[18]5*6)(3|[29]5*6)*(5|[29]5*[18])|(3|[18]5*[07]|(2|9|[18]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))(6|35*[18]|(4|35*6)(3|[29]5*6)*(5|[29]5*[18])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))*(0|7|35*[29]|(4|35*6)(3|[29]5*6)*(6|[29]5*[29])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(3|65*[29]|(0|7|65*6)(3|[29]5*6)*(6|[29]5*[29]))))(4|[07]5*[29]|(1|8|[07]5*6)(3|[29]5*6)*(6|[29]5*[29])|(2|9|[07]5*[07]|(1|8|[07]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(3|65*[29]|(0|7|65*6)(3|[29]5*6)*(6|[29]5*[29]))|(3|[07]5*[18]|(1|8|[07]5*6)(3|[29]5*6)*(5|[29]5*[18])|(2|9|[07]5*[07]|(1|8|[07]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))(6|35*[18]|(4|35*6)(3|[29]5*6)*(5|[29]5*[18])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))*(0|7|35*[29]|(4|35*6)(3|[29]5*6)*(6|[29]5*[29])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(3|65*[29]|(0|7|65*6)(3|[29]5*6)*(6|[29]5*[29]))))*(5|[07]5*3|(1|8|[07]5*6)(3|[29]5*6)*(0|7|[29]5*3)|(2|9|[07]5*[07]|(1|8|[07]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(4|65*3|(0|7|65*6)(3|[29]5*6)*(0|7|[29]5*3))|(3|[07]5*[18]|(1|8|[07]5*6)(3|[29]5*6)*(5|[29]5*[18])|(2|9|[07]5*[07]|(1|8|[07]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))(6|35*[18]|(4|35*6)(3|[29]5*6)*(5|[29]5*[18])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))*(1|8|35*3|(4|35*6)(3|[29]5*6)*(0|7|[29]5*3)|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(4|65*3|(0|7|65*6)(3|[29]5*6)*(0|7|[29]5*3)))))(2|9|45*3|(5|45*6)(3|[29]5*6)*(0|7|[29]5*3)|(6|45*[07]|(5|45*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(4|65*3|(0|7|65*6)(3|[29]5*6)*(0|7|[29]5*3))|(0|7|45*[18]|(5|45*6)(3|[29]5*6)*(5|[29]5*[18])|(6|45*[07]|(5|45*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))(6|35*[18]|(4|35*6)(3|[29]5*6)*(5|[29]5*[18])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))*(1|8|35*3|(4|35*6)(3|[29]5*6)*(0|7|[29]5*3)|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(4|65*3|(0|7|65*6)(3|[29]5*6)*(0|7|[29]5*3)))|(1|8|45*[29]|(5|45*6)(3|[29]5*6)*(6|[29]5*[29])|(6|45*[07]|(5|45*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(3|65*[29]|(0|7|65*6)(3|[29]5*6)*(6|[29]5*[29]))|(0|7|45*[18]|(5|45*6)(3|[29]5*6)*(5|[29]5*[18])|(6|45*[07]|(5|45*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))(6|35*[18]|(4|35*6)(3|[29]5*6)*(5|[29]5*[18])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))*(0|7|35*[29]|(4|35*6)(3|[29]5*6)*(6|[29]5*[29])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(3|65*[29]|(0|7|65*6)(3|[29]5*6)*(6|[29]5*[29]))))(4|[07]5*[29]|(1|8|[07]5*6)(3|[29]5*6)*(6|[29]5*[29])|(2|9|[07]5*[07]|(1|8|[07]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(3|65*[29]|(0|7|65*6)(3|[29]5*6)*(6|[29]5*[29]))|(3|[07]5*[18]|(1|8|[07]5*6)(3|[29]5*6)*(5|[29]5*[18])|(2|9|[07]5*[07]|(1|8|[07]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))(6|35*[18]|(4|35*6)(3|[29]5*6)*(5|[29]5*[18])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))*(0|7|35*[29]|(4|35*6)(3|[29]5*6)*(6|[29]5*[29])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(3|65*[29]|(0|7|65*6)(3|[29]5*6)*(6|[29]5*[29]))))*(5|[07]5*3|(1|8|[07]5*6)(3|[29]5*6)*(0|7|[29]5*3)|(2|9|[07]5*[07]|(1|8|[07]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(4|65*3|(0|7|65*6)(3|[29]5*6)*(0|7|[29]5*3))|(3|[07]5*[18]|(1|8|[07]5*6)(3|[29]5*6)*(5|[29]5*[18])|(2|9|[07]5*[07]|(1|8|[07]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))(6|35*[18]|(4|35*6)(3|[29]5*6)*(5|[29]5*[18])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))*(1|8|35*3|(4|35*6)(3|[29]5*6)*(0|7|[29]5*3)|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(4|65*3|(0|7|65*6)(3|[29]5*6)*(0|7|[29]5*3)))))*(3|45*4|(5|45*6)(3|[29]5*6)*(1|8|[29]5*4)|(6|45*[07]|(5|45*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(5|65*4|(0|7|65*6)(3|[29]5*6)*(1|8|[29]5*4))|(0|7|45*[18]|(5|45*6)(3|[29]5*6)*(5|[29]5*[18])|(6|45*[07]|(5|45*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))(6|35*[18]|(4|35*6)(3|[29]5*6)*(5|[29]5*[18])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))*(2|9|35*4|(4|35*6)(3|[29]5*6)*(1|8|[29]5*4)|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(5|65*4|(0|7|65*6)(3|[29]5*6)*(1|8|[29]5*4)))|(1|8|45*[29]|(5|45*6)(3|[29]5*6)*(6|[29]5*[29])|(6|45*[07]|(5|45*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(3|65*[29]|(0|7|65*6)(3|[29]5*6)*(6|[29]5*[29]))|(0|7|45*[18]|(5|45*6)(3|[29]5*6)*(5|[29]5*[18])|(6|45*[07]|(5|45*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))(6|35*[18]|(4|35*6)(3|[29]5*6)*(5|[29]5*[18])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))*(0|7|35*[29]|(4|35*6)(3|[29]5*6)*(6|[29]5*[29])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(3|65*[29]|(0|7|65*6)(3|[29]5*6)*(6|[29]5*[29]))))(4|[07]5*[29]|(1|8|[07]5*6)(3|[29]5*6)*(6|[29]5*[29])|(2|9|[07]5*[07]|(1|8|[07]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(3|65*[29]|(0|7|65*6)(3|[29]5*6)*(6|[29]5*[29]))|(3|[07]5*[18]|(1|8|[07]5*6)(3|[29]5*6)*(5|[29]5*[18])|(2|9|[07]5*[07]|(1|8|[07]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))(6|35*[18]|(4|35*6)(3|[29]5*6)*(5|[29]5*[18])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))*(0|7|35*[29]|(4|35*6)(3|[29]5*6)*(6|[29]5*[29])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(3|65*[29]|(0|7|65*6)(3|[29]5*6)*(6|[29]5*[29]))))*(6|[07]5*4|(1|8|[07]5*6)(3|[29]5*6)*(1|8|[29]5*4)|(2|9|[07]5*[07]|(1|8|[07]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(5|65*4|(0|7|65*6)(3|[29]5*6)*(1|8|[29]5*4))|(3|[07]5*[18]|(1|8|[07]5*6)(3|[29]5*6)*(5|[29]5*[18])|(2|9|[07]5*[07]|(1|8|[07]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))(6|35*[18]|(4|35*6)(3|[29]5*6)*(5|[29]5*[18])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))*(2|9|35*4|(4|35*6)(3|[29]5*6)*(1|8|[29]5*4)|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(5|65*4|(0|7|65*6)(3|[29]5*6)*(1|8|[29]5*4))))))*$

which is “only” 12,733 characters, and works fine in the implementations I’ve tried.

---

PS. To see how much less interesting it is using recursion, here’s a Perl program that tests for divisibility by 7 using a recursive regular expression:

my $re = qr{
  ^($|[07](?1)|[18](?2)|[29](?3)|3(?4)|4(?5)|5(?6)|6(?7))
  |(?!)(?:
      ([07](?4)|[18](?5)|[29](?6)|3(?7)|4(?1)|5(?2)|6(?3))
    | ([07](?7)|[18](?1)|[29](?2)|3(?3)|4(?4)|5(?5)|6(?6))
    | ([07](?3)|[18](?4)|[29](?5)|3(?6)|4(?7)|5(?1)|6(?2))
    | ([07](?6)|[18](?7)|[29](?1)|3(?2)|4(?3)|5(?4)|6(?5))
    | ([07](?2)|[18](?3)|[29](?4)|3(?5)|4(?6)|5(?7)|6(?1))
    | ([07](?5)|[18](?6)|[29](?7)|3(?1)|4(?2)|5(?3)|6(?4))
  )
}x;

while (<>) {
    print(/$re/ ? "matches\n" : "does not match\n");
}

This is just a direct expression of the DFA as a recursive regular expression.