It's often more convenient in practice to use feet and yards than meters because you can divide them evenly into three, four, six, or nine equal units, with a convenient whole number of inches. For the same reason, the Babylonian sexagesimal system remains in use today for angles (360 degrees, each subdivided into 60 minutes of 60 seconds). In some sense, both 10 and 12 are composite numbers, but 12 seems much "more composite", and 60 is "more composite" still, much more so than 100.
What are the "most composite" numbers, the ones that can be evenly divided in the most different ways? Of course you can always get more factors if you go to a larger number, but some numbers have more factors than any number smaller than themselves; there's a definite sequence of these "most composite" numbers, and there aren't really very many of them. You might think that you'd just go with the products of the primes: 2, 2·3 = 6, 2·3·5 = 30, 2·3·5·7 = 210, and so on; but it turns out that 30 only has six proper divisors greater than 1 (2, 3, 5, 6, 10, and 15), but 24 beats it by having the same number of factors and being smaller. It substitutes a versatile, lightweight extra couple of 2s for 30's 5, thus getting the same number of factors sooner.
It turns out our old friends 12, 36, 60, and 360 are in this sequence, along with 4 and 6. However, 360·60 = 21600 is not, because once you get past 720, it behooves you to start throwing a 7 in there. The champion that beats poor old Babylonian 21600, with its mere 70 divisors, is 10080 (2⁵·3²·5·7) with 70 divisors, or alternatively 20160 (2⁶·3²·5·7) with a whopping 82 divisors! (I'm still leaving out 1 and the number itself, here.)
Here's what I've computed so far:
>>> nfactors = lambda n: len([x for x in range(2, n) if n % x == 0])
>>> i = 1
>>> best_nfactors = 0
>>> while True:
... i += 1
... n = nfactors(i)
... if n > best_nfactors:
... print i, n
... best_nfactors = n
...
4 1
6 2
12 4
24 6
36 7
48 8
60 10
120 14
180 16
240 18
360 22
720 28
840 30
1260 34
1680 38
2520 46
5040 58
7560 62
10080 70
15120 78
20160 82
25200 88
27720 94
45360 98
50400 106
55440 118
83160 126
110880 142
Unsurprisingly, I'm not the first person to discover this; the standard term is "highly composite number", and it's sequence A002182 in OEIS.
If, on the other hand, you're looking for a base for a positional numbering system that will give you maximum divisibility, 30 beats 24, and 210 is better still. True enough, in base 24, an eighth is simply 0.3, while in base 30, it's 0.3,22,15, which is more awkward; but in base 30, a fifth is 0.6, while in base 24, it's an infinite sequence: 0.4,0,19,4,0,19,...! Infinite sequences are far more awkward to calculate with. Base 210 gives you perfect divisibility --- in a finite number of fractional positions, but maybe more than one --- for any number of portions up to 10, and any composite number up to 21, plus of course many other numbers.