[Verse 1]
Two numbers standing, need to find their bond
The greatest divisor they both respond
Start with the larger, call it A today
Smaller one's B, now watch the display
Divide A by B, grab that remainder
This ancient method, proven game-changer
Euclid discovered twenty-three centuries back
Mathematical treasure, staying on track
[Chorus]
Divide and conquer, swap and repeat
Until that remainder hits zero complete
GCD emerges from this ancient dance
Euclidean rhythm, give it a chance
Modulo magic, reducing the size
Greatest common factor before your eyes
[Verse 2]
Example brewing: forty-eight meets eighteen
Forty-eight mod eighteen equals twelve clean
Now eighteen becomes the dividend new
Eighteen mod twelve gives us six, it's true
Twelve mod six lands us zero at last
Six is our answer, algorithm passed
Recursive beauty in every iteration
Efficient solution across every nation
[Chorus]
Divide and conquer, swap and repeat
Until that remainder hits zero complete
GCD emerges from this ancient dance
Euclidean rhythm, give it a chance
Modulo magic, reducing the size
Greatest common factor before your eyes
[Bridge]
When remainder vanishes into the void
Last divisor standing can't be destroyed
That's your GCD, the treasure you sought
Computational elegance, perfectly wrought
Time complexity logarithmic and tight
Mathematical precision burning bright
[Verse 3]
Code it recursive or iterative style
Both implementations worth your while
While B not zero, keep the cycle alive
A becomes B, then B gets the modulo dive
Extended version gives you more than just GCD
Coefficients dancing in linear harmony
Bezout equation solved with elegant grace
Euclidean algorithm conquers every case
[Chorus]
Divide and conquer, swap and repeat
Until that remainder hits zero complete
GCD emerges from this ancient dance
Euclidean rhythm, give it a chance
Modulo magic, reducing the size
Greatest common factor before your eyes
[Outro]
From cryptography to fraction reduction
Euclidean power drives construction
Ancient wisdom meets modern computation
GCD mastery across every nation