Base 12 has an easy test for divisibility by 2, 3, 6, and so forth: just look at the last (few) digit(s). Base 10 likewise has an easy test for divisibility by 2, 4, 5, etc. Bases 8 and 16 give only rules for 2, 4, 8, and so on, more restricted than both the preceding.Factors and Prime Factors: octal has two factors, one prime; decimal and two factors, both prime; dozenal has four factors, two prime.
We must also consider that binary multiplication and division is the simplest and most easily attempted form of all; easier than tenery or decimal or any other kind.
We've also to consider that arrangement by six is actually the most efficient method;
That sixty-four - the square of eight- is the first cubic and square number;
That twelve is the first abundant number;
And a host of other things (hopefully this has gotten the ball rolling...)
Base 12 has an easy test for divisibility by 11: sum the digits and test for divisibility by 11. Base 10 has a similar rule for divisibility by 3 and 9. Base 8's rule is for 7. Base 16 gets 3 and 5.
Base 12 has an easy rule for divisibility by 13: sum alternating digits and test for divisibility by 13. (12A -> A - 2 + 1 = 9, so 12A is not divisible by 13.) Base 10's rule is for 11, and base 8 gets 9 (and so also 3). Base 16 gets 17, which is less useful (but overlooked, so maybe good).
Base 1000 (and so base 10 by extension) has a great rule for divisibility by 7, 11, and 13: sum alternating digits (since 11*13*7 = 1001). I don't know of a similar rule for bases 8, 12, or 16 -- but this might be worth exploring.
Base 10 has a good binary conversion: 2^10 ~= 10^3. Hexadecimal has a similar rule, but not quite as nice: 16^5 ~= 10^6. The others don't have anything obvious.