[Verse 1]
When coefficients dance with unknowns in a line
A times x plus b times y equals c divine
But integers demand their special place to be
There's a secret gatekeeper holding the key
[Chorus]
Greatest common divisor must divide the right side clean
If it doesn't split evenly, no solutions can be seen
But when it does divide through, infinite answers flow
Linear Diophantine secrets that every student should know
[Verse 2]
Find one solution first, call it x-naught and y-naught
Then shift by multiples, that's the pattern you've caught
Add b over d times t to your starting x coordinate
Subtract a over d times t from y, don't deviate
[Chorus]
Greatest common divisor must divide the right side clean
If it doesn't split evenly, no solutions can be seen
But when it does divide through, infinite answers flow
Linear Diophantine secrets that every student should know
[Bridge]
Parameter t can be any integer you choose
Positive, negative, zero - you simply cannot lose
The spacing stays consistent, solutions march in rows
Like soldiers in formation, each step the pattern shows
[Verse 3]
Bezout's identity lurks behind this theorem's face
Extended Euclidean helps you find that starting place
From concrete to abstract, integers hold their ground
In Diophantine's kingdom where whole number solutions are found
[Chorus]
Greatest common divisor must divide the right side clean
If it doesn't split evenly, no solutions can be seen
But when it does divide through, infinite answers flow
Linear Diophantine secrets that every student should know
[Outro]
When gcd divides c, solutions multiply
Integer coordinates beneath the algebraic sky