Alberto Monteiro wrote:
>
> David Hobby wrote:
> >
> >> The problem with base 12 is that it has _2_ twice and _3_ once
> >> when you factor it, so that the "practical man" rules to check
> >> if a number is divisible by another would get a higher degree
> >> of confusion. Base 6 would be a much better choice than base 12.
> >
> > I'm not sure what you mean.
> >
> The rules for "power of 2" and "power of 3" would be different in
> base 12: we would have a last-digit test for 2 and 4, a two-last-digits
> test for 8 and 16, etc. In base 10, we have a _n_-last-digits test
> for 2^n and 5^n. Also, when a number ends in zeroes, we have just
> to count them to know the lesser power of 2 and 5, in base 12, we
> would get the power of 4 and 3.
Correct. But I still don't see why it's much harder.
The natural thing to expect is that each number should have a
separate divisibility test, anyway.
...
> >> I don't see many advantages in base 6 over base 10:
> >> the only one that comes to my mind is that base 10 has simple
> >> rules to check if a number is divisible by 2, 5, 3, 9 and 11;
> >
> > I think the rules for 4,6 and 8 are also simple. (Again, here's
> > a link for background: http://www.jimloy.com/number/divis.htm )
> >
> 4 and 6 are simple, but not so much. The rule for 8 is, IMHO, horrible.
Let's agree that the rule for 4 is usually easy for anyone
who knows their multiplication tables (out to 8x12, I guess). You
look at the last two digits, and see if they form a multiple of 8
or are 4 more than a multiple of 8.
Now for the rule for 8. It splits into two cases. If the
100s digit is even, the last two digits must form a multiple of 8.
If the 100s digit is odd, the last two digits must form 4 more than
a multiple of 8.
...
> > (This is making base 6 look good. But
> > there should be a way to lift divisibility rules from base 6 to
> > base 12 (=2*6), at the price of adding some complexity.)
...
Upon reflection, there is no useful way to get base 12
rules from base 6 rules for divisibility by 5 or by 7. So I
claim that base 6 is best. (Good rules for 2,3,6, O.K. rules
for 4,5,9 fair rules for 7,8,10, and awful rules for the rest.)
...
> > This is fuzzy, as I said. I
> > would count the base 10 rule for 3 as much less simple than the
> > base 10 rule for 8, even. I guess it depends on what size
> > numbers one is expecting to use the divisibility tests on--
> > I'm imagining large numbers as input.
> >
> I am imagining _small_ numbers, those that 5-th grade students
> are required to factor.
>
> Alberto Monteiro
But if the numbers are small, why not just divide?
Division is relatively painless if you don't have to keep
track of what the quotient is, but only care about the
remainder.
---David
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