="Warren Smith" <warren.wds@gmail.com>
If you add and subtract alternately the 6-bit blocks
Of course this isn't strictly binary, but rather uses "fat digits" encoded in base 2. Anything interesting happen if the blocks aren't the same size? If they overlap? Adding digits and/or alternating can be generalized by accumulating products scaled by some sequence of small constants besides +1 and -1. Different sequences can be used to produce a suite of divisibility tests (and IIRC this has been discussed on this list anon). One way I've used to test quickly for divisibility by a bunch of small divisors: multiply enough of them together to get P just shy of the word size, then use binary gcd(N,P). You can of course use a table of P's. Another hack I've pulled in microcode on awkward architectures is multiply by approximate fixed-point reciprocals "normalized" to lie in 0.5..1.0. Sometimes the result can be shown to be close enough to detect by an easy test, or whacked with Newton's method (like the famous square root bum).
Related question to "Can we make fast divisibility tests for binary computers?" is "...and should anybody really want to?"
I dunno, but on this list that's a bit like responding to G. H. Hardy's famous riposte by asking "And should anyone really want to make babies?"