I was inspired by this Reddit thread to see if I could supplement the traditional regex primality test with a regex Euclidean algorithm. Here's the shortest one that occurred to me; it takes the two numbers (assumed to be positive integers) whose GCD we want to compute in $a and $b: 
(1 x $a) =~ /(?:$1)*(.*)/ and ($a, $b) = ($b, length $1) while $b

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At the end, $a is the GCD (and $b is 0). Life's even better if we're allowed to start with $a and $b strings of 1s of the desired length, and leave our answer in $a in the same form: 
($a, $b) = ($b, $a =~ /(?:$1)*(.*)/) while $b

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In reply to Find the GCD regexically by JadeNB

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