I was wondering about residue number systems and I was thinking that in particular bases that are close to a power of 2 are relatively easy to reduce modulo. Nathan Laredo came up with the list 63 62 61 59 55 53. (* 63 62 61 59 55 53) = 40978178010. 68 719 476 736 is the number of possible 36-bit numbers; 40 978 178 010 is a reasonably large fraction of that. All the fractional bits of loss between all those don't add up to even a whole bit of loss!
So consider, like, 1411 and 8675309. 1411 % [63 62 61 59 55 53] is [25 47 8 54 36 33]; 8675309 reduces to [20 21 11 8 49 4]. If we multiply these elementwise we get [ 500 987 88 432 1764 132], which reduces to [59 57 27 19 4 26]. And waves hands with the Chinese Remainder Theorem we can get the product of 1411 * 8675309 = 12240860999, modulo 40978178010 anyway.
This works for addition and subtraction; you can only get division if the bases are actually prime instead of relatively prime I think.
The reason this is potentially interesting is that six circuits to multiply two six-bit numbers modulo a six-bit base are a lot cheaper than one circuit to multiply 34-bit numbers, and have a lower path length to boot, so you can clock them faster. So RNSs like this get used a lot for high-sample-rate DSP.