Can ax + by + cz = d have an integer solution?

  • Context: Undergrad 
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Discussion Overview

The discussion revolves around the conditions under which the Diophantine equation ax + by + cz = d has integer solutions. It includes aspects of proofs, mathematical reasoning, and clarification of the equation's structure.

Discussion Character

  • Homework-related, Mathematical reasoning, Conceptual clarification

Main Points Raised

  • One participant asks for help proving that the equation ax + by + cz = d has an integer solution if and only if gcd(a, b, c) divides d.
  • Another participant suggests starting with a simpler equation, ax + d, and adding terms incrementally to understand the problem.
  • A later post corrects an earlier mistake, clarifying that the equation is ax + by + cz = d, which prompts a discussion about the implications of this correction.
  • One participant references the Linear Equation Theorem, stating that the equation ax + by = gcd(a, b) always has integer solutions, which can be found using the Euclidean algorithm.
  • Another participant expresses that they figured out the problem after receiving help.
  • There is a clarification that the original misstatement of ax + by + cz + d is not an equation, which highlights the importance of precise notation in mathematical discussions.

Areas of Agreement / Disagreement

The discussion includes some corrections and clarifications, but there is no consensus on the proof or the conditions for integer solutions as participants explore different aspects of the problem.

Contextual Notes

Participants express uncertainty regarding the implications of the equation's structure and the conditions under which integer solutions exist. The discussion does not resolve these uncertainties.

ACardAttack
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Prove that the Diophantine equation ax+by+cz+d has an integer solution if and only if the gcd(a,b,c) divides d.

Got this on my homework for my proofs class. Help would be greatly appreciated.

Thanks
 
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Start with a simpler equation, ax+d, and then try to add the other terms one by one.
 
Dodo said:
Start with a simpler equation, ax+d, and then try to add the other terms one by one.

I goofed...it is actually ax+by+cz=d...does that make a difference?
 
the Linear Equation Theorem says that the equation ax + by = gcd(a, b) always has a solution(s, u) in integers, and this solution can be found by the Euclidean algorithm, which we use to compute the gcd of a and b.
 
I figured it out...thanks for the help
 
ACardAttack said:
I goofed...it is actually ax+by+cz=d...does that make a difference?
Well, yes! It's an equation! ax+ by+ cz+ d isn't an equation.
 

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