# Sum of unknowns

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• May 13th 2009, 11:48 AM
Aquafina
Sum of unknowns
I've edited the original question because there was a mistake in it.

can x and y be determined if it is given that there are 65 values which the sum a multiple of x and a multiple of y cannot give

I.e. Nx + My has 65 unattainable values, where N and M are variables. Also x and y both do not equal 2.

I saw it on: Hard math questions, again? - Yahoo! Answers

Its the last part to the last question.
• May 13th 2009, 12:40 PM
e^(i*pi)
Quote:

Originally Posted by Aquafina
can x and y be determined if it is given that a multiple of x and a multiple of y added together give 65.

I.e. Nx + My = 65, where N and M are integers. Also x and y both do not equal 2.

No, you need another equation
• May 13th 2009, 02:25 PM
I-Think
Diophantine equation
Salutations.

A question to e^i*pi, if the values of M and N are given, couldn't you solve this as a Diophantine equation?

Question (Not relevant) #2: Why does spell-check insist I capitalize 'Diophantine'?
• May 13th 2009, 08:07 PM
Isomorphism
Quote:

Originally Posted by I-Think
Salutations.

A question to e^i*pi, if the values of M and N are given, couldn't you solve this as a Diophantine equation?

Yes, you could based on a simple divisibility condition on the gcd(M,N).

Quote:

Question (Not relevant) #2: Why does spell-check insist I capitalize 'Diophantine'?
Because its named after Diophantus,I guess....
• May 13th 2009, 08:53 PM
Aquafina
Thank you, no M and N are not given. Any reason why it doesn't work without it?
• May 13th 2009, 09:00 PM
Isomorphism
Quote:

Originally Posted by Aquafina
Thank you, no M and N are not given. Any reason why it doesn't work without it?

For such a general case read the "linear diophantine equation" section here.
• May 13th 2009, 09:25 PM
Aquafina
hi thanks, so the reasoning is we can't solve it because we don't know if c is the greatest common divisor of a and b?

But i thought that is just the case if we want to have infinite solutions?

i.e. with pentagonal square numbers http://mathworld.wolfram.com/images/...dEquation5.gif would have infinite integer solutions, right?

Pentagonal Square Number -- from Wolfram MathWorld
• May 13th 2009, 10:04 PM
Isomorphism
Quote:

Originally Posted by Aquafina
hi thanks, so the reasoning is we can't solve it because we don't know if c is the greatest common divisor of a and b?

But i thought that is just the case if we want to have infinite solutions?

It suffices if gcd(M,N) divides 65 to have infinite solutions, else it wont have any solution...

To see this observe that if gcd(M,N) does not divide 65 and there exists a solution, then gcd(M,N) divides the left hand side, but not the right hand side...

If it does divide, then use division algorithm to solve for $\displaystyle Mx+Ny = \text{gcd}(M,N)$. Let $\displaystyle (x_0,y_0)$ be one solution and then generate $\displaystyle x = x_0 + Nt, y = y_0 - Mt$ for various t, to get infinite solutions.
• May 13th 2009, 10:11 PM
Aquafina
ah thanks, so that would imply all diophantine equations either have infinite integer solutions or no solutions?

or is this just for linear ones? i.e. the example i used, x^2 - 6y^2 = 1 is not a correct example
• May 13th 2009, 10:15 PM
Isomorphism
Quote:

Originally Posted by Aquafina
ah thanks, so that would imply all diophantine equations either have infinite integer solutions or no solutions?

or is this just for linear ones? i.e. the example i used, x^2 - 6y^2 = 1 is not a correct example

Its definitely true for all linear ones. You cant say it in general :(

Note: The one you quoted is of the form called "Pell's equation". Read the wiki on it if you are interested...
• May 13th 2009, 10:37 PM
Aquafina
Ah right, I think I'll make a new thread for the Pell's Equations then lol

Anyways, did u just edit your previous post because this wasnt there before (i think)

To see this observe that if gcd(M,N) does not divide 65 and there exists a solution, then gcd(M,N) divides the left hand side, but not the right hand side...

If it does divide, then use division algorithm to solve for http://www.mathhelpforum.com/math-he...2ce905ac-1.gif. Let http://www.mathhelpforum.com/math-he...8e672c3e-1.gif be one solution and then generate http://www.mathhelpforum.com/math-he...eacabf18-1.gif for various t, to get infinite solutions.

I havent done algorithms (sorry http://www.mathhelpforum.com/math-he...cons/icon9.gif) could you explain that paragraph if you dont mind?
• May 13th 2009, 11:00 PM
Aquafina
Error in original question
oops i just realised im doing the question wrong...

It is actually:

can x and y be determined if it is given that there are 65 values which the sum a multiple of x and a multiple of y cannot give

I.e. Nx + My has 65 unattainable values, where N and M are variables. Also x and y both do not equal 2.

I saw it on: Hard math questions, again? - Yahoo! Answers

Its the last part to the last question.
• May 13th 2009, 11:12 PM
Isomorphism
Quote:

Originally Posted by Aquafina
Ah right, I think I'll make a new thread for the Pell's Equations then lol

Anyways, did u just edit your previous post because this wasnt there before (i think)

Yes

Quote:

I havent done algorithms (sorry ) could you explain that paragraph if you dont mind?
Dont worry about the name.. Its the method you use to compute the gcd..
Do you know Bezout's identity?
• May 13th 2009, 11:28 PM
Aquafina
read the post above, question slightly modified
• May 14th 2009, 01:18 AM
Aquafina
chicken mcnugget theorem..
I think this requires the chicken mcnugget theorem

but i'm not sure how to use it. any ideas?

Chicken McNugget Theorem - AoPSWiki
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