Discussion Overview
The discussion revolves around the solvability of various linear Diophantine equations, specifically examining conditions under which integer solutions exist. Participants analyze specific equations and explore methods for determining their solvability, including the use of greatest common divisors (gcd) and modular arithmetic.
Discussion Character
- Technical explanation
- Mathematical reasoning
- Debate/contested
Main Points Raised
- Some participants assert that if \(a\) and \(b\) are relatively prime, then \(ax + by = N\) has integer solutions for any integer \(N\), suggesting that statement (a) is true.
- One participant calculates \(\gcd(70, 42) = 14\) and concludes that for equation (b), since 14 does not divide 1409, there are no integer solutions.
- In contrast, the same participant states that for equation (c), since 14 divides 1428, there are integer solutions.
- For equation (d), it is noted that \(\gcd(2016, 4031) = 1\) which divides the right-hand side, indicating that integer solutions exist.
- Another participant echoes the previous claims about equations (b), (c), and (d), reinforcing the conclusions drawn about their solvability.
- Participants inquire about quicker methods for solving these equations, with one suggesting the use of modular arithmetic, particularly noting that for equation (b), the left-hand side is even while the right-hand side is odd, which implies no solutions.
Areas of Agreement / Disagreement
Participants generally agree on the analysis of equations (b), (c), and (d), but there is no explicit consensus on the validity of statement (a) as some participants have not confirmed their agreement. The discussion remains open regarding the quickest methods for solving these types of equations.
Contextual Notes
Participants rely on the properties of gcd and modular arithmetic, but the discussion does not resolve the broader implications of these methods or their applicability to all linear Diophantine equations.