gustavo toja
gtojafrachia@aol.com

Should anything pops up in my mind, I'll post it here.

>I didnīt try with 13 yet.

Then I claim the glory, see

https://www.cut-the-knot.org/blue/div7-11-13.shtml

I am very glad you used my method to improve it and generalize it.
YOu have this glory yes!
Thank you!

See you soon. I am working in another surprising thing about this method.

I am looking for your further contributions.

Best regards,
Alex

Best regards

Gustavo Toja

Please keep your site up. Let me know if you decide to discontinue it, as a link at my site will become dead.

Thank you,
Alex

Best regards

Gustavo

Alex

I updated my site. Please take a look.
Thank you.

Gustavo

Thank you for the link. I have also updated my page.

Let's have a look at one of your examples: 342,563,182,319.
Below n will indicate the number of pairs less 1.

34 25 63 18 23 19 (n = 5)
6 4 0 4 2 5
1 4 7 4 5 5
554741
55 47 41 (n = 2)
6 5 6
6 2 6
626
6 26 (n = 1)
6 5
1 5
51

The remainder of 51 is 2.
N = the sums of n = 5 + 2 + 1 = 8.
10⁸ = 2 (mod 7).

I claim that

2 × 342,563,182,319 = 2 (mod 7).

Therefore, 342,563,182,319 = 1 (mod 7).

Is that clear enough?

Gustavo

If you ask me, I would not at all teach division by 7 in school, middle or high, to an "average" student, but only to those who show interest.

I argue thus

1. The ability to find out, fast or slow, whether a number is divisible by 7 is of no value in and of itself.
   
2. Without understanding of why the thing works, it's of no value. Neither of the proofs, yours or mine, will be successfuly grasped by an "average" 10th, let alone 7th grader.
   

I just ran across the following article

https://209.157.64.200/focus/f-news/1380269/posts

The article is too angry in my view, although I do agree with the basics.

I do not think that a question whether a topic is suitable to a certain grade is good. Students are too different and must be let be.

I deeply respect your opinion.
I taught math for 22 years and I am a psychologist. I know that one of the things that a teacher (or psychologist) has to do is to respect the individual and to let him develop his abilities with no external limits. I don't think that grades or labels are fundamental things but I believe that they are important as references. I donīt know (I also recognize that I don't know most of things)any successful educational experience that gets a human being significantly better than tradicional education. I deeply know experiences like the KUMON Method, which theaches math without labeling children ( a 2nd-grade student can learn algebra if he's able to do so or a 7th-grade student studies sum + 2 if he doesn't have this skill) or the brilliant theories of Robert M. Gagné.
I believe that children have to know math as a tool and if something is important for their development they have to use it, even they don't know why it works. He will have time to learn later. Example, my son (who is 5 years-old) uses scissors and has no idea of their physical principles, he counts and he doesn't know anything about Set Theory, maybe he only has an intuitive idea. By the way, most of things that we learn in our first twelve years will be re-meant (i. e. will get the real meaning) after (sometimes many years after or never, like understanding women). I would teach that the circumference length is twice the ratio times pi, before I explain transcendental numbers. It's only a tool and it could be important for him. In the same way, I would show a rainbow to my son because it's beautiful even if he is not able to understand why it appears.
One of the reasons that I like mathematics a lot is the fact that it's surprising. It is funny. I want to marvel a 7th- grade math student with a new method of divisibility by seven, not because it will be important for his life, but because maybe he will be interested in knowing more about maths. I wonīt say at this moment why this method works because it's (probably) out of his comprehension but I will let him curious and maybe he will wish to know more. Maybe I can explain why it works for a 4-digit number (I have a 7th-grade son who understood the demonstration for 4-digit numbers).
I know that my opinion is not shared for most of modern educators and I am not trying to convince anybody that I am right. By the way, I wonder who knows what is right in Education.

Gustavo Toja

P.S. My english is very limited. When I have to express my opinion or an idea I can be unkind without intention. Sorry if I was.

>I deeply respect your opinion.

As I do yours.

>I don't think that
>grades or labels are fundamental things but I believe that
>they are important as references.

I do not disagree with you, a caveat being that the question whether anything is suitable for a 7th grader I interpret as it beig suitable for an average student of a certain age or for a classroom presentation. With this interpretation, I say "No", I would not even try. Moreover, I do not see a reason why one should.

>I believe that children have to know math as a tool

Which may be quite true. If so, what they can do with a criteria for divisibility by 7?

>and if
>something is important for their development they have to
>use it, even they don't know why it works.

To paraphrase an idea of R. P. Boath, by practicing in the (n+1)st grade student learn what they were supposed to have learned in the nth grade. If that's true, it appears that in the nth grade students going through the actions with learning what they were supposed to. So I am not sure that in such a generality your assessment is correct.

>He will have time
>to learn later. Example, my son (who is 5 years-old) uses
>scissors and has no idea of their physical principles, he
>counts and he doesn't know anything about Set Theory, maybe
>he only has an intuitive idea.

First, I absolutely have no objections, nor comprehensions of teaching anything to an individual student. You may want to check a story at

https://www.cut-the-knot.org/pythagoras/12MatchesAreaPuzzle.shtml

It's about a relative of mine who is now 10.

>By the way, most of things
>that we learn in our first twelve years will be re-meant (i.
>e. will get the real meaning) after (sometimes many years
>after or never, like understanding women).

You mean "... and we still marry them, hmm ..."?

>I would teach
>that the circumference length is twice the ratio times pi,
>before I explain transcendental numbers. Itīs only a tool
>and it could be important for him.

That's fine. What do you say about

https://www.cut-the-knot.org/pythagoras/NatureOfPi.shtml

>In the same way, I would
>show a rainbow to my son because it's beautiful even if he
>is not able to understand why it appears.

Why not to teach something that he can understand? Surely there is plenty of this variety too.

>One of the reasons that I like mathematics a lot is the fact
>that it's surprising. It is funny.

Agree.

>I want to marvel a 7th-
>grade math student with a new method of divisibility by
>seven, not because it will be important for his life, but
>because maybe he will be interested in knowing more about
>maths.

Sure. Try it, why not? But only with individual students.

>P.S. My english is very limited. When I have to express my
>opinion or an idea I can be unkind without intention. Sorry
>if I was.

I had an uneasy feeling that my terse reply might have irked you. I did not mean that. I love mathematics. I loved it as far back as I can remember. I am sorry for people who do not have a clue about its beauty. But I know for sure that most of the attempts to prozelitize fail misrerably. Have a look at the Guest book section of this forum. I do not know how much abuse I am getting.

All the best,
Alexander

Thanks for your quick and smart reply. I am learning many things with you.
Well, you are afraid of your reply might have irked me and I am afraid of my reply might have been unkind. So, we are both wrong!
This happens because we are not talking sat at a table, with a bottle of beer in the middle (or vodka if you prefer).
The links you recommended were very instructives. I liked them so much.
I am exploring your page to discover more interesting things!

Thank you again.

Gustavo

This is certainly an unfortunate circumstance.

>(or vodka if you prefer).

Nope, but beer would do.

Gustavo, if I may ask you. I am closing this thread. For, it became to unwieldy. If and when you want to take this up again, please start a new thread.

Gustavo

Suppose we have a number: 3437

Split it in groups of 3 from the right: 3 and 437

Now subtract the lhs from the rhs: 437-3 = 434.

Once I got to this stage used to divide and find out.....2 steps that takes but we can also use Alex or Gustavos method from here on and find out if the number is divisible by 7,11 or 13.

If it is a larger number: 38391787 ..... the one gustavo used in his example..

After the first step: 38 , 391 , 787

Second step: 787 - 391 + 38 = 434

Note the alternate - and + sign... and then we divide actually to find out....

Actually its not very elegant with big numbers as we have to do 3 digit'subtractions and additions.....but it is an alternate method and its nice :)

Hope this helps someone somewhere.

I saw this clever divisibility trick from a link on Science News magazine's website (https://www.sciencenews.org), and I thought I would work out the general background. Since I am not familiar with making HTML pages that show mathematical formulas, I wrote up my results using LaTex and made it into a PDF file, which I placed on my website.

It describes the general method for constructing factoring tricks, but it doesn't talk specifically about these new tricks that you have been discussing here. Please have a look at it here:

https://home.comcast.net/~stuartmanderson

This has been a very interesting discussion, and I am happy if I am able to contribute something to it.

Let me know what you think.

Stuart Anderson

I am reading carefully your paper and it'seems to be really interesting. When I finish I will send my comments to you. Thanks for your comments about the Toja's Method.

Gustavo

Nice to know. I missed that column.

>It describes the general method for constructing factoring
>tricks, but it doesn't talk specifically about these new
>tricks that you have been discussing here. Please have a
>look at it here:
>
>https://home.comcast.net/~stuartmanderson

This is very good. The derivation aside, it does fall into a general approach to divisibility described at

https://www.cut-the-knot.org/blue/divisibility.shtml

What you do can be simplified. This is the Horner scheme for computing values of polynomials. In this method, you replace the repeated factor 10² by its remainder of division by 7, which is 2. So that

((a·10² + b)·10² + c)·10² ... = ((a·2 + b)·2 + c)·2 ... (mod 7)

I believe I saw this somewhere, but do not remember where.

This said, finding specific examples for the general theory pointed to by the above link requires a lot of ingenuity. I am grateful for having a chance to learn of your example. The approach is certainly interesting. For divisibility by 11, it leads to an extremely simple result: split the number into 2 digit terms and sum them up. If the result is divisible by 11, so is the original number.

I am going to upgrade my pages and make a link to your site. Please let me know if you decide to discontinue it.

>This has been a very interesting discussion, and I am happy
>if I am able to contribute something to it.

Many thanks
Alex

>I am going to upgrade my pages and make a link to your site.

While doing just that I noticed an example f₅ on

https://www.cut-the-knot.org/blue/divisibility.shtml

that is exactly this criteria of divisibility by 11. It's awful: I should probably take a rest.

>This is very good. The derivation aside, it does fall into a
>general approach to divisibility described at
>
>https://www.cut-the-knot.org/blue/divisibility.shtml

Yes, I was looking at this just yesterday. While I was making a general
framework for deriving rules, you were making a general scheme for
classifying them. These are very compatible endeavors, I think.
>
>What you do can be simplified. This is the Horner scheme for
>computing values of polynomials. In this method, you replace
>the repeated factor 10² by its remainder of
>division by 7, which is 2. So that
>
>((a·10² + b)·10² + c)·10²
>... = ((a·2 + b)·2 + c)·2 ... (mod 7)
>
I see what you mean about polynomial computation. This is the method that was recommended for computing polynomials in the manual of the first Hewlett Packard calculator I owned when I was starting college, because these calculators use Reverse Polish Notation, which makes this computation particularly easy.

You are certainly correct that what I wrote could be simplified. I deliberately wrote the article using many concrete examples, so that it could be easily followed by people whose familiarity with mathematical notation is limited. Also, I was attempting to retrace the thought process that I went through, because it is more interesting (to me, at least) to read about the path that was taken, rather than just the final result. The little summary at the end really contains all anyone needs to know to actually use the method.

>I believe I saw this somewhere, but do not remember where.
>
>This said, finding specific examples for the general theory
>pointed to by the above link requires a lot of ingenuity. I
>am grateful for having a chance to learn of your example.
>The approach is certainly interesting. For divisibility by
>11, it leads to an extremely simple result: split the number
>into 2 digit terms and sum them up. If the result is
>divisible by 11, so is the original number.
>
>I am going to upgrade my pages and make a link to your site.
>Please let me know if you decide to discontinue it.

I will be leaving this page up permanently (as far as I can predict), but I will be sure to inform you if I have to take it down for some reason. By the way, I will be adding some other articles soon, perhaps later today. I have a collection of miscellaneous mathematical results that I did for fun over the years, and I'm just now starting to put them on my site.

>
>>This has been a very interesting discussion, and I am happy
>>if I am able to contribute something to it.
>
>Many thanks
>Alex

Thank you for your very interesting website.

Stuart

This is from Vorobjev's classic (referenced on the page) I learned years ago as a school boy.

>These are very compatible endeavors, I
>think.

Of course.

>I see what you mean about polynomial computation. This is
>the method that was recommended for computing polynomials in
>the manual of the first Hewlett Packard calculator I owned
>when I was starting college

No wonder. It is standard in Numerical Analysis courses.

>I deliberately wrote the article using many
>concrete examples, so that it could be easily followed by
>people whose familiarity with mathematical notation is
>limited.

It was an easy and enloyable reading.

>By the way, I will be adding some
>other articles soon, perhaps later today. I have a
>collection of miscellaneous mathematical results that I did
>for fun over the years, and I'm just now starting to put
>them on my site.

Please keep us informed.

>Thank you for your very interesting website.

Thank you,
Alex