This question/examples sounds like what we used to do in 3rd
and 4th grade at one school I attended. We called the
process, "casting out nines." We were required to use this
method to check our arithmetic work (I suspect it was also
to improve a arithmetic skills). The method worked like
this:
Add the following numbers: 2637 + 3815 = 6452
Now apply the "casting out nines" check. Add the digits of
each number together. While adding, if you equal or exceed
9, subtract 9 and continue.
(2+6)+3 = 11, caste out nine
11-9=2, continue
2+7=9, caste out nine
9-9=0
So, next to 2637 we write 0.
For 3815, we would write 8.
For 6452, we would write 8.
Our check, then, is 0 + 8 = 8.
We did this for all our arithmetics homework and classwork.
-- John R GannCollege of Micronesia - FSM PO Box 159 Pohnpei, FM 96941 AOL Instant Messenger: Mathletics Homepage: http://ourworld.compuserve.com/homepages/mathlete/home.htm
"Tough guys don't do math. Tough guys fry chicken for a living." - Jaime Escalante (as portrayed in "Stand and Deliver")
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Received: from archives.math.utk.edu - 128.169.244.106 by mail.fm with Microsoft SMTPSVC; Wed, 10 Feb 1999 13:56:07 +1100 Received: by archives.math.utk.edu (8.8.5/2.10math-UTK) id UAA00416; Tue, 9 Feb 1999 20:31:05 -0500 (EST) Received: from kcmetro.cc.mo.us by archives.math.utk.edu with ESMTP (8.8.5/2.10math-UTK) id UAA00411; Tue, 9 Feb 1999 20:30:58 -0500 (EST) Received: from haehl ([165.173.21.114]) by kcmetro.cc.mo.us (AIX4.2/UCB 8.7/8.7) with SMTP id TAA386304; Tue, 9 Feb 1999 19:43:42 -0600 (CST) Message-ID: <004401be548e$254ac960$7215ada5@haehl> From: "Martha Haehl" <haehl@KCMETRO.CC.MO.US> To: <mathedcc@archives.math.utk.edu> Cc: <brownb@maplewoods.cc.mo.us> Subject: [MATHEDCC] Number theory question Date: Tue, 9 Feb 1999 19:40:59 -0500 MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 4.72.2106.4 X-MimeOLE: Produced By Microsoft MimeOLE V4.72.2106.4 Sender: owner-mathedcc@archives.math.utk.edu Precedence: bulk Reply-To: "Martha Haehl" <haehl@KCMETRO.CC.MO.US> Return-Path: owner-mathedcc@archives.math.utk.edu
Last night, over dinner with friends, one friend, Sam told me about how he "reduces a number to its number." This was a very non-mathematical description of a pattern he has recognized with three digit numbers. I am curious if anyone has studied this pattern or proved its general truth. It goes something like this:
Being a mathematician, I will define a digit function where the function of a 3-digit number is defined as follows. 1. Add the digits. If the sum is a single digit, that digit is the output. 2. If the sum is 2 digits, add the digits again and that digit is the output.
For example f(769) = 4 because 7 + 6 + 9 = 22 and 2 + 2 = 4.
The interesting number theory connection is that if you take the 3 digits and form any combination of a 2 digit number and a 1 digit number and add the numbers, then add the digits (more than once if necessary), the final output digit will be the same.
For example, take the digits 7, 6, and 9 to form 79 + 6 = 85. 8 + 5 = 13, 1 + 3 = 4, giving f(769).
Likewise, 96 + 7 = 103, 1 + 0 + 3 = 4 which is also f(769).
I have experimented with several 3-digit numbers and in all cases, the sum of any 2-digit number and the remaining digit map (as indicated above) to f(number) as defined above.
I have also experimented with 4-digit numbers and various combinations of 3 and 1-digit numbers added and mapped, then 2-digit and 2-digit numbers from the 4-digits mapped to the sum of the 4-digits mapped to a single digit as above. In every case, I find that the mappings all lead to the same single digit. Does anyone know of such a pattern that has been studied in number theory? Is there a proof that such a phenomenon holds for n-digit numbers?
By the way, on grading, in Basic Math this semester, my students are getting a 0 grade for any quiz/test less than 80%. Before they can take a retake, they must get the signature from a tutor in the math lab that they have gone through the test and worked more problems like the ones they missed. So far, students have grumbled a bit, but have gotten help. I have required retakes before, but this is the first time I have required students to get help from someone besides me before taking a retake. This is making a big difference in the time I spend on the class.
Martha
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