I've been lurking on this discussion, since I have a sort of vested
interest in the result of the discussion, as will become apparent shortly...
As many people noted, the rule for 7 is to strike off the last digit of the
number, then subtract twice the digit you struck off from the number which
was left after you struck it off. There are other rules for divisibility
by all the other prime numbers (for 11, form the alternating sum of the
digits; for 13, add 4 times the last digit to the remaining number; for 17,
subtract 5 times the last digit, and so on).
You can turbocharge all of this, and make it easier by noting that any
sequence of digits in the number can be replaced by the remainder upon
division by p, then checking that. For instance, is 988 divisible by 19?
Replacing 98 by its remainder upon division by 19, which is 03 (i.e., 3),
you get 038=38, which is twice 19, so the answer is yes.
Some of the divisibility rules are available on a faq from the dr. math
website, but the rules given there are, generally, more complicated than
necessary.
In fact, once you know that the number is divisible by p, it is *really
easy* to actually do the division in your head, and faster than on a
calculator, at least when the number is less than 100p.
And now for my "vested interest." I wrote a paper on this subject over the
summer, entitled "Divisibility rules and mental division," which I just
sent off to The Mathematics Teacher. I would be glad to send a copy of it
to anyone who requests it (regrettably, it has a lot of PICT graphics in it
because of my equation editor, so I will have to send it to you by snail
mail). Send requests to msnyder@fsc.edu, or to msnyder@tiac.net. I will
also be giving a talk on this at Fitchburg State later this month, so if
you are in north central Mass, and would like to attend, let me know.
One of my fears, of course, is that in fact some paper has already been
published on this subject. I was unable to find any, but if you know of
any, please let me know (although you may hear of my doing my "Samurai
Mathematician" bit afterwards...)
Mark A. Snyder Work: 978.665.3076
Dept. of Mathematics Home: 978.386.7158
Fitchburg State College email: msnyder@tiac.net,
Fitchburg, Mass. 01420 msnyder@fsc.edu
fax: 978.665.4031
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