11.
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(Programming Fundamentals) Diophantine Equations**
For a linear Diophantine equation, ax + by = c. What is the value of x, if a solution exists? (Where a, b, c are not zeros and g = gcd(a,b) )
Pick ONE option
Show answer & explanation
Answer: D. X = (c/g) * (b/g)^-1 mod (a/g)
The linear Diophantine equation ax + by = c has integer solutions if and only if g = gcd(a, b) divides c. The particular solution for x is derived using the extended Euclidean algorithm: first find g and coefficients such that a*(a/g) + b*(b/g) ≡ 1 (mod perspective), then x ≡ (c/g) * (b/g)^-1 (mod a/g), where (b/g)^-1 is the modular multiplicative inverse of (b/g) modulo (a/g).
Step-by-step Derivation:
Step-by-step derivation:
- For ax + by = c to have integer solutions, g = gcd(a,b) must divide c.
- Divide the equation by g: (a/g)x + (b/g)y = c/g.
- Let a' = a/g, b' = b/g, c' = c/g. Now gcd(a', b') = 1 (coprime).
- We need a'x + b'y = c', which means a'x ≡ c' (mod b').
- Solving for x: x ≡ c' * (a')^-1 (mod b'), equivalently x ≡ (c/g) * (a/g)^-1 (mod b/g).
- However, by rearranging to b'y ≡ -a'x (mod a'), we get y ≡ -c' * (b')^-1 (mod a'), and x ≡ (c/g) * (b/g)^-1 (mod a/g).
- The standard form used in computational number theory for finding one particular solution is Option D.