Number theory problem

[Obelix]

How many numbers lie between 11 and 1111 which when divided by 9 leave a remainder of 6 and when divided by 21 leave a remainder of 12?

[DrRocket]

How many numbers lie between 11 and 1111 which when divided by 9 leave a remainder of 6 and when divided by 21 leave a remainder of 12?

 

18

 

On how many bulletin boards have you posted the same question ?

http://www.thescienceforum.com/Number-theory-problem-30858t.php

[Obelix]

How many numbers lie between 11 and 1111 which when divided by 9 leave a remainder of 6 and when divided by 21 leave a remainder of 12?

18

 

Kindly justify your answer.

 

On how many bulletin boards have you posted the same question ?

http://www.thescienceforum.com/Number-theory-problem-30858t.php

 

On two - 2 -

I found it here: PHORUM. Is anything wrong with multiple posts? I'm collecting different solutions.

Edited May 19, 2011 by Obelix

[DrRocket]

Kindly justify your answer.

 

 

 

On two - 2 -

I found it here: PHORUM. Is anything wrong with multiple posts? I'm collecting different solutions.

 

Figure it out for yourself.

 

Multiple posting is not good form.

[Obelix]

Figure it out for yourself.

Multiple posting is not good form.

 

1) Kindly define "good" and "bad form".

 

2) You too do multiple (and a bit impolite!) posting: The Science forum.

[DrRocket]

1) Kindly define "good" and "bad form".

 

You should be able to figure this out for yourself also.

 

2) You too do multiple (and a bit impolite!) posting: The Science forum.

 

I post answers, not multiple questions. Big difference.

[Hal.]

How many numbers lie between 11 and 1111 which when divided by 9 leave a remainder of 6 and when divided by 21 leave a remainder of 12?

 

I think that the meaning of the words ' between 11 and 1111 ' should be made clear .

I think the meaning in english can be , 11 to 1111 , inclusive or exclusive of 11 and 1111.

 

Also the word ' and ' can have possible different uses where , 1 , it means the numbers which satisfy both conditions ,

2 , it means the numbers which satisfy either the first or the second conditions .

[DrRocket]

How many numbers lie between 11 and 1111 which when divided by 9 leave a remainder of 6 and when divided by 21 leave a remainder of 12?

 

I think that the meaning of the words ' between 11 and 1111 ' should be made clear .

I think the meaning in english can be , 11 to 1111 , inclusive or exclusive of 11 and 1111.

 

Also the word ' and ' can have possible different uses where , 1 , it means the numbers which satisfy both conditions ,

2 , it means the numbers which satisfy either the first or the second conditions .

 

"Between 11 and 1111" is clear and in this case it does not matter whether one means inclusive or exclusive.

 

"and" is well-understood in mathematics. It means "and". When one means "or", one says "or". Pretty simple, no ?

[Hal.]

I'll give an example of the use of the word ' and ' which shows an alternative meaning which is what I am taught to see if I can and include it as a possibility . This example is the same type as that of the original post . So , now there are two .

 

In the group of numbers from 1 to 10 inclusive , Write down the numbers which are less than 3 and greater than 8 .

Using one meaning of the word ' and ' I would write down no numbers . Using another meaning of the word ' and ' I would write down 1,2,9,10

 

Inclusive or Exclusive doesn't matter in the case of the original post , but that is a different point than the meaning taken , when taking one meaning of ' between ' as compared with taking another . What I'm saying is that I'll consider my math to include both meanings when I see the word ' between ' used , if it is unclear to me which to take .

[DJBruce]

In the group of numbers from 1 to 10 inclusive , Write down the numbers which are less than 3 and greater than 8 .

Using one meaning of the word ' and ' I would write down no numbers . Using another meaning of the word ' and ' I would write down 1,2,9,10

 

 

No, the word and in the technical sense of mathematics means the intersection of two events or sets. So in your example you would have:

[math]\{x<3 | x \in R\}[/math] and [math]\{x>8 | x \in R\}[/math]

 

[math]\{x<3 | x \in R\} \cap \{x>8 | x \in R\}=\{\}[/math]

 

The second interpretation you gave is actually the interpretation of "or" ie: the union of two sets.

[Hal.]

So , DJBruce , the first example you give is consistent with one of my meanings , unless there is something I don't see .

 

So , on I'll proceed to the second meaning . You say I give a meaning to ' and ' which is described by ' or ' . I know what you mean and I would follow this rule if that was clear to me mathmatically . But , the questions of the original poster and my example are not in math language . They are in english . That is the point of what I say , when I say that there is another meaning for ' and ' .

 

These questions are not put to those being asked in a form that would be seen in Electronics , Computer Programming etc , which is why I see the alternative . It is a point of view , a different interpretation of ' and ' .

Edited May 19, 2011 by Hal.

[DrRocket]

 

In the group of numbers from 1 to 10 inclusive , Write down the numbers which are less than 3 and greater than 8 .

..... . Using another meaning of the word ' and ' I would write down 1,2,9,10

 

 

And you would be wrong.

[Hal.]

Define the rules to make the conclusion . I defined mine for mine .

[DJBruce]

So , DJBruce , the first example you give is consistent with one of my meanings , unless there is something I don't see .

Correct.

 

So , on I'll proceed to the second meaning . You say I give a meaning to ' and ' which is described by ' or ' . I know what you mean and I would follow this rule if that was clear to me mathmatically . But , the questions of the original poster and my example are not in math language . They are in english . That is the point of what I say , when I say that there is another meaning for ' and ' .

 

These questions are not put to those being asked in a form that would be seen in Electronics , Computer Programming etc , which is why I see the alternative . It is a point of view , a different interpretation of ' and ' .

 

Mathematics has developed a language all of its own that includes words, which have very precise and technical mathematical definitions. "And" and "Or" are two of these words, which when read by a mathematician in a post relating to mathematics would instantly mean something very precise to the reader. I believe this is what DrRocket said, '"and" is well-understood in mathematics.'

 

This technical language is nothing special to mathematics, as almost all sciences have a very well developed and precise "language". For example, the word force has an everyday definition, but if one were to read "force" in a physics article one would instantly know that they are more than likely referring to the technical definition. Also the technical definitions of "and" and "or" is definitely used in computer science and physics.

Edited May 19, 2011 by DJBruce

[DrRocket]

Define the rules to make the conclusion . I defined mine for mine .

 

You are certainly free to live in your own little world. But your personal definition is worthless in the world of mathematics, and the initial quetion was mathematical in nature.

[imatfaal]

Is this an interesting question? - the answer is clearly 18

 

the numbers are 33 96 159 222 285 348 411 474 537 600 663 726 789 852 915 978 1041 1104. A few minutes looking at it and you can realise that the coincidence has to repeat every 63 - and that 33 is the lowest possible, a quick check on excel and its shown. I guess that given a few sleepless nights I could work out how to do this for any set of original numbers (or maybe I couldn't) - but it just seems boring.

 

I am just curious - is there anything noteworthy about this?

[DJBruce]

I am just curious - is there anything noteworthy about this?

 

Pretty much just an application of a modified version of the Chinese Remainder Theorem, where your moduli are not pairwise coprime, which I guess is pretty cool in a way.

 

 

See about 1/3 from the bottom of the "Theorem Statement" section.

http://en.wikipedia....mainder_theorem

Edited May 19, 2011 by DJBruce

[Hal.]

DJBruce , my views include your views and a linguistic variant non standard mathmatically in nature with a mathmatical interpretation for the purpose of exploration of alternatives .

 

not posted in order .

 

Dr.Rocket , my little world is the same little world that is the little world of your little world .

 

My mathmatical world is not the same mathmatical world as the mathmatical world that is your mathmatical world .

Edited May 19, 2011 by Hal.

[imatfaal]

Thanks for that DJ - will try and get my head around it. Seeing the names brahmagupta and fibonacci being bandied around in the article - i think my estimate of a couple of nights to generalise might have been a little ambitious.

[DrRocket]

Is this an interesting question? - the answer is clearly 18

 

the numbers are 33 96 159 222 285 348 411 474 537 600 663 726 789 852 915 978 1041 1104. A few minutes looking at it and you can realise that the coincidence has to repeat every 63 - and that 33 is the lowest possible, a quick check on excel and its shown. I guess that given a few sleepless nights I could work out how to do this for any set of original numbers (or maybe I couldn't) - but it just seems boring.

 

I am just curious - is there anything noteworthy about this?

 

 

Your list is correct. But you don't have to actually calculate the numbers to see how many of them exist:

 

 

1. Between 11 and 1111 there are 53 numbers that are congruent to 12 mod 21. They are of the form 21k+12 , k=0,1,...,52

 

2. The question is how many of them are also congruent to 6 mod 9; i.e. for which k is there an m so that 21k+12=9m+6

 

3. Now 21k+12 = 3k+3 mod 9 = 3(k-1) + 6 mod 9 So 21k + 12 = 6 mod 9 if and only if (k-1) is a multiple of 3.

 

4. From the list in #1 those numbers are 1, 4, ..., 52. There are 18 of them.

[Hal.]

Dr.Rocket , you said my personal definition is worthless in the world of mathmatics . It's not . You fail to understand how my view is equally as reasonable as any upon consideration . This , doubtless , is all I can expect .

[DrRocket]

Dr.Rocket , you said my personal definition is worthless in the world of mathmatics . It's not . You fail to understand how my view is equally as reasonable as any upon consideration . This , doubtless , is all I can expect .

 

It is a matter of communication and accepted definition. Choosing an unconventional meaning for standard terminology , without making a very visible exception to convention, is not reasonable.

 

In other words, your personal definition is worse than worthless. It cultivates confusion.

Edited May 20, 2011 by DrRocket

[Hal.]

I explained my meaning clearly from my first post , thus , you make less sense than previously .

 

 

[DJBruce]

Thanks for that DJ - will try and get my head around it. Seeing the names brahmagupta and fibonacci being bandied around in the article - i think my estimate of a couple of nights to generalise might have been a little ambitious.

 

Proving the generalization to any numbers might be more difficult, but actually doing the problem or generalizing without proof shouldn't be horrible. The question asks for numbers of the form:

 

[math]x \equiv 6mod(9)[/math]

[math]x \equiv 12mod(21)[/math]

 

So this is similar to the Chinese Remainder Theorem, but since [math]gcd(9,21)=3\neq1[/math] we cannot you the apply the regular chinese remainder theorem, but

 

 

[math]6 \equiv 12mod(gcd(21,9))[/math]

 

So we can apply the special case where we know that we want to look at numbers congruent to the lcd(21,9)=63. So this tells us that our ring has 63 equivalence classes, and so every 63 number must meet the desired conditions of the problem. You can fairly easily find that 33 is the smallest number that meets the desired criteria, and so then we see that between 33 and 1111 we have 17 multiples of 63, and so you now that you have a total of 18 between 11 and 1111.

 

This process should generalize to certain other situations fairly easily, but I'll let you work that out if you want.