[Verse 1]
When you have two numbers that don't share a prime
And two equations running at the same time
X equals A when divided by M
X equals B when divided by N
There's a magic theorem from ancient days
That shows us there's a unique solution always
[Chorus]
Chinese Remainder breaks it down
One big problem, split around
When the moduli are coprime friends
The system solves, the story ends
Unique solution, that's the key
Modulo M times N, you see
Chinese Remainder shows the way
Independent parts at play
[Verse 2]
Greatest common divisor must be one
That's when the real magic has begun
The integers split into separate rings
Z mod MN becomes two smaller things
Isomorphic structures, hand in hand
Breaking complex into what we understand
[Chorus]
Chinese Remainder breaks it down
One big problem, split around
When the moduli are coprime friends
The system solves, the story ends
Unique solution, that's the key
Modulo M times N, you see
Chinese Remainder shows the way
Independent parts at play
[Bridge]
Pairwise coprime, extend the chain
M one, M two, up to M K again
Each congruence standing on its own
Product of moduli sets the zone
Algebraic structure decomposed
First glimpse of how math is composed
[Verse 3]
From ancient Chinese mathematicians wise
To modern algebra before our eyes
One system becomes a product space
Each component finds its rightful place
The foundation stone of number theory
Making complex problems less scary
[Chorus]
Chinese Remainder breaks it down
One big problem, split around
When the moduli are coprime friends
The system solves, the story ends
Unique solution, that's the key
Modulo M times N, you see
Chinese Remainder shows the way
Independent parts at play
[Outro]
When GCD is one, the path is clear
Chinese Remainder Theorem is here
Split the problem, find the answer true
Ancient wisdom still carries us through