> One of our staff in a math lab recently created a summary of
> divisibility rules for dividing by the whole numbers 2,3,...,10.
>
> example: A number is divisible by 2 if its ones digit is 0,2,4,6,8.
>
> It states there is no rule for 7.
> -----------------------------------------------------------------
> Assuming the domain of discussion to be the whole numbers, is
> this true?
No, this is NOT true. There are (must be) divisibility rules for
_every_ integer divisor. The big question is whether the rules for
a particular divisor are simple enough to use practically or
to compete with direct division. The case of 7 is kind of borderline
in this respect.
Because the remainder on dividing 10 by 7 is 3, a "triple forward"
method works. Thus on meeting 14 you can think: take the 1,
triple it, and add it to 4 to get a result of 7, which gives you an
answer of YES. Using this method on 105 you first get 35, then
14, then 7 again.
Because the remainder on dividing 100 by 7 is 2, you can use a
"double and twice forward" method, too. In this case the 1 from 105
would get doubled and then added directly to the 5 to give you 7 in
one step. With both of these methods you should "cast out 7's",
reducing 9's to 2's, etc. More generally, you should cast out _any_
combination of digits that is a multiple of 7 (visual patterning).
For example, the number 2114049 is clearly divisible by 7 as it
stands, right?
Using either of the methods above (or direct division), it can be
shown that 1001 is a multiple of 7. This means that, in theory,
we can reduce any multidigit number to 1000 or less by simply
adding and subtracting alternate groups of three digits before
checking for divisibility by 7 (works for 13 and 11, too). In
practice, this is often too tough to carry out. But sometimes
opportunity strikes. For example, consider 6111. Subtract 6
from 111 to get 105, and take it from there.
It is also possible to perform a divisibility test by working
backward from the _end_ of a multidigit number. The appropriate
multiplier to use for going back one or two places depends on
the divisor you are working with. In the case of 7 it is
"quintuple back", of course casting out 7's if possible.
Thus for 14, take 4 times 5 and add to 1 to get 21. Then take
1 times 5 and add to 2 to get 7. Bingo! Silly in this case,
of course, but sometimes it works well. Let's see, for 30002
we get 3010 --> 301 --> 35 --> 28 --> 21 etc. (stop when you
know the answer!). The multiplier for "twice back" is... well,
I'll let you find it. :-)}
As you can easily see, these ideas can be used to develop divisibility
tests for other numbers, too. The point is that there are a lot more
divisibility tests than just the familiar ones that get repeated in all
the books, and exploring this topic with students can be rewarding.
Another twist on this is developing divisibility tests in other bases.
All this coming from a calculator freak like me? Just reinforces
what I have said many times, that to use a calculator properly
you need to be _more_ nimble with number patterns and
number sense than you do if you have a pencil and can just push
those digits around on paper. I spend a lot of time in my classes
trying to develop this attitude in my students.
RWW Taylor
National Technical Institute for the Deaf
Rochester Institute of Technology
Rochester NY 14623
>>>> The plural of mongoose begins with p. <<<<
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