I've found this to be INTERESTING..

ZHero · 2010-02-17 19:09:28

Find all pairs of Integers (X,Y) such that..
X*Y is congruent to X+Y mod 9

(0,0) and (8,5) is what i can see by trial..
can anyone SHOW THE SOLUTION?
thx

Last edited by ZHero (2010-02-17 20:49:01)

bobbym · 2010-02-17 19:26:31

Hi ZHero;

Neither of those examples are solutions, 0^0 does not leave a remainder of 0 when divided by nine. 8^5 is congruent to 8 mod 9 not (8+5).

The smallest positive solution I could find was x = 513 and y = 675 there are others.

ZHero · 2010-02-17 20:44:29

Hi bobbym..
X*Y is NOT X^Y
its X x Y or X multiplied by Y

I'll edit my previous post to Capitalize x and y so that it looks Better. Sorry for the inconvenience caused by not using LaTex!
:-)

Last edited by ZHero (2010-02-17 20:47:49)

bobbym · 2010-02-17 21:01:48

Hi ZHero;

No problem my vision is that bad.

First: all x,y that are divisible by 9 are solutions.

Ricky · 2010-02-18 02:08:47

First: all x,y that are divisible by 9 are solutions.

This is what ZHero meant by (0,0). Anything congruent to a solution will of course be a solution.

ZHero, the way you typically do these problems is you eliminate groups of cases till there is a small enough number to check by hand. For example, (x, 0) and (0, y) will not be solutions for x and y nonzero. Further, there are no solutions for x=1 or y=1.

No solutions where x+y = 9 except for x=3 and y=6 (or vice versa), and this is a solution.

John E. Franklin · 2010-02-18 05:35:55

for (8,5)
8 times 5 is 40 and 8 + 5 = 13.
40 - 36 = 4 and 13 - 9 = 4

Last edited by John E. Franklin (2010-02-18 05:36:43)

ZHero · 2010-02-18 05:54:05

John E. Franklin wrote:

for (8,5)
8 times 5 is 40 and 8 + 5 = 13.
40 - 36 = 4 and 13 - 9 = 4

yes.. (8,5) is a solution! U just substituted the values in place of (X,Y) whereas i wanna know HOW TO GET THESE VALUES!?

bobbym · 2010-02-18 06:22:13

Hi guys;

Sorry for the confusion I caused by not seeing the problem correctly ( I thought it was x^y ).

Normally for a small mod like that I would just plow through the 64 possibilities of

x = 1,2,3,...8

y =1,2,3,...8

The solutions then are:

(2,2),(3,6),(5,8),(6,3),(8,5)

Of course I have left out (0,0) as trivial and you already have that pair.

Ricky · 2010-02-18 09:35:41

Normally for a small mod like that I would just plow through the 64 possibilities of

My post reduced it to 15 possibilities.

bobbym · 2010-02-18 12:46:54

Yes, but it required thinking. In the time it takes to reason about it my co-worker hands in the answer. He gets the raise and I am fired for being slow. Computing has replaced thinking.

Principle of Expediency: A good answer today is better than a great answer tomorrow.

Ricky · 2010-02-18 14:00:24

Computing has replaced thinking.

Let's see you go compute all groups of order 8633 up to isomorphism.

Principle of Expediency: A good answer today is better than a great answer tomorrow.

Why are both not an option?

In any case, I'll wager the time it took for me to find those results was much shorter than the time it would take you to write a program.

mathsyperson · 2010-02-18 19:32:21

Maybe, but both are pretty quick.

for (int i = 0; i <= 9 ; i++)
{
     for (int j = i; j <= 9; j++)
     {
          if((i+j) % 9 == (i*j) % 9)
          {
               Console.WriteLine("(" + i + "," + j + ")")
          }
     }
}

That was around 2 minutes.

bobbym · 2010-02-19 03:20:59

Hi

Normally for a small mod like that I would just plow through the 64 possibilities

Ricky wrote:

Let's see you go compute all groups of order 8633 up to isomorphism.

That's why I clearly said for small mod.

Why are both not an option?

They both are: Just don't think about which one for too long because then you are violating P.O.E.

In any case, I'll wager the time it took for me to find those results was much shorter than the time it would take you to write a program.

Table[{x,y,Mod[x*y,9]==Mod[x+y,9]},{x,1,8},{y,1,8}]//Column

1 minute to write it:
1 minute to inspect the output:

Computing has replaced thinking.

Sorry, not being precise here. Human thinking is what I meant.

ZHero · 2010-02-19 17:51:36

X*Y-(X+Y)=9K
(X-1)(Y-1)=9K+1
for k=0, 1, 2 etc..
for k=0
(X-1)(Y-1)=1=1*1=-1*-1
X=2 Y=2 or X=0 Y=0 and so on....

bobbym · 2010-02-20 00:48:53

Hi ZHero;

That idea unless I am missing something will not find (6,3) or (3,6) for any integer k.

Ricky · 2010-02-20 04:35:15

That's why I clearly said for small mod.

No, you said "Computing has replaced thinking."

bobbym · 2010-02-20 05:08:57

Hi Ricky;

Yes, for this problem it has. No one has offered a general solution to this, so it seems suited to computer assault. Supposing I said mod 127. All I would have to do to recompute is to change two constants. You would have to rethink the entire problem. If I said x - y rather than x + y you only have to change a plus to a minus and your done.

A beginning programmer with Basic would know enough to form 2 loops and solve this. That is why no thinking is necessary here.

Ricky · 2010-02-20 06:28:41

Ok, I'll accept that you were talking about in this problem only. However, by "replace", did you mean that there is no use for thinking?

bobbym · 2010-02-20 08:20:34

I don't know, Rick. I think it is on the way out. Will anyone miss it?

I think you missed the obvious reason why I didn't employ any thinking on this problem. And that's really incorrect, programming is thinking. You did it already in post #5! I didn't have anything to add to that.

Ricky · 2010-02-21 02:05:44

I think it is on the way out. Will anyone miss it?

And once again you sound like you're talking about in general. And even if you only meant it referring to this problem (I don't see how...), it's a ridiculous statement to make.

And that's really incorrect, programming is thinking.

Were you really of the opinion that I thought a brain dead person could program? Obviously there is less thinking when one uses a brute force method, that's why it's called brute force. This is the difference we're talking about.

phrontister · 2010-02-21 04:11:30

I hope I understand this problem correctly. Anyway, here's what I've found:

Solutions occur when (x+y) mod 9 is either 4 or zero.

Last edited by phrontister (2010-02-22 00:48:22)

bobbym · 2010-02-21 04:27:55

Hi Rick;

And once again you sound like you're talking about in general. And even if you only meant it referring to this problem (I don't see how...), it's a ridiculous statement to make.

You just can't pull out one little statement in a post. You have to read it all.

Not so ridiculous. Just a perception an opinion really. You have to have been around long enough to have seen the difference, Machines went from a 17 operations per second clunker to Teraflops. Do you think that kind of change left man unaffected?

Were you really of the opinion that I thought a brain dead person could program?

No. I was refuting my own statement that no thinking was required to write those programs.

I hate commenting about a phrase such as brain dead. Let me just say many of the people who were higher up than I ( status, rank or intelligence ), shared that exact opinion and voiced it often.

Ricky · 2010-02-21 06:31:32

I'm not really sure I understand much of your post. I have no idea why the recent trends in computing power have anything to do with the discussion of computation vs. thought.

In any case, you don't wait for a problem to be uncomputable to start learning how to think about problems. For example, my method of eliminating obvious choices is exactly how one goes about classifying simple Lie algebras. Learning a method means you can attempt to apply that method to other problems.

Seeing a problem solved in various ways can't hurt you.

LQ · 2010-02-21 06:49:18

The hurt lies in how you came to the point of solving the problem which we also feel. Definitely endureable ricky, but definitely an aouch. How ever you solved it, it was finite, and hence I am here, good bye and thank you.

bobbym · 2010-02-21 08:00:59

Hi;

I'm not really sure I understand much of your post. I have no idea why the recent trends in computing power have anything to do with the discussion of computation vs. thought.

It has everything to do with it. The faster machines get the more brute force algorithms are used.. More and more problems are handled by putting them between loops. Take the factoring of large numbers for instance. Currently it is in the realm of topology and number theory. You need all kinds of tough math to factor say a 150 - 200 digit number. It would take months or years. But if you have a quantum computer you can factor that in a microsecond. Using trial division, the stupidest algorithm in the world. Hardware replaces thinking because we humans will always do what is easiest and fastest.

In any case, you don't wait for a problem to be uncomputable to start learning how to think about problems.

You don't use a cannon to kill a mosquito. I have already explained why I answered that problem like that. I explained it candidly in post #19. This is what I wrote.

bobbym wrote:

You did it already in post #5! I didn't have anything to add to that.

Had you provided the solutions instead of just pruning the tree, I would not even have posted anything, and none of this would be happening. Simply, there was nothing more to be said on top of your analysis. So I finished up the question using a computer method. I did not use your ideas in the body of my program because frankly it would take too long. The dumb program was the fastest and most reliable way to now get the actual answers.

Seeing a problem solved in various ways can't hurt you.

Now this one is incomprehensible. Who said it could? Methods, I am famous for bizarre methods of solving problems. Almost anyone can come up with a simpler way to do every problem I solve. If that problem would have been too tough for a brute force approach I would have made every attempt to reduce the number of possibilities.

Math Is Fun Forum

#1 2010-02-17 19:09:28

I've found this to be INTERESTING..

#2 2010-02-17 19:26:31

Re: I've found this to be INTERESTING..

#3 2010-02-17 20:44:29

Re: I've found this to be INTERESTING..

#4 2010-02-17 21:01:48

Re: I've found this to be INTERESTING..

#5 2010-02-18 02:08:47

Re: I've found this to be INTERESTING..

#6 2010-02-18 05:35:55

Re: I've found this to be INTERESTING..

#7 2010-02-18 05:54:05

Re: I've found this to be INTERESTING..

#8 2010-02-18 06:22:13

Re: I've found this to be INTERESTING..

#9 2010-02-18 09:35:41

Re: I've found this to be INTERESTING..

#10 2010-02-18 12:46:54

Re: I've found this to be INTERESTING..

#11 2010-02-18 14:00:24

Re: I've found this to be INTERESTING..

#12 2010-02-18 19:32:21

Re: I've found this to be INTERESTING..

#13 2010-02-19 03:20:59

Re: I've found this to be INTERESTING..

#14 2010-02-19 17:51:36

Re: I've found this to be INTERESTING..

#15 2010-02-20 00:48:53

Re: I've found this to be INTERESTING..

#16 2010-02-20 04:35:15

Re: I've found this to be INTERESTING..

#17 2010-02-20 05:08:57

Re: I've found this to be INTERESTING..

#18 2010-02-20 06:28:41

Re: I've found this to be INTERESTING..

#19 2010-02-20 08:20:34

Re: I've found this to be INTERESTING..

#20 2010-02-21 02:05:44

Re: I've found this to be INTERESTING..

#21 2010-02-21 04:11:30

Re: I've found this to be INTERESTING..

#22 2010-02-21 04:27:55

Re: I've found this to be INTERESTING..

#23 2010-02-21 06:31:32

Re: I've found this to be INTERESTING..

#24 2010-02-21 06:49:18

Re: I've found this to be INTERESTING..

#25 2010-02-21 08:00:59

Re: I've found this to be INTERESTING..

Board footer