[Verse 1]
When numbers dance in modular space
There's a special bond they share
If a equals b in mod n's embrace
Then patterns show up everywhere
It's reflexive like a mirror's face
Symmetric when we flip and swap
Transitive chains that interlace
These properties will never stop
[Chorus]
Add them up, they stay the same
Multiply, it's still the game
Power up to any height
Congruence keeps the balance right
But division's not so clean
Greatest common divisor's the key
One is what we need to see
For the rule to guarantee
[Verse 2]
If a congruent b and c congruent d
Both dancing to mod n's beat
Then a plus c and b plus d
Will make the pattern complete
Times tables work the same way too
When congruence leads the dance
Every operation follows through
Given just the right circumstance
[Chorus]
Add them up, they stay the same
Multiply, it's still the game
Power up to any height
Congruence keeps the balance right
But division's not so clean
Greatest common divisor's the key
One is what we need to see
For the rule to guarantee
[Bridge]
Watch out for the division trap
When ac equals bc mod n
Don't assume that a equals b
Check the gcd my friend
If c and n share common ground
The rule might break apart
But when their gcd equals one
Division works from the start
[Chorus]
Add them up, they stay the same
Multiply, it's still the game
Power up to any height
Congruence keeps the balance right
But division's not so clean
Greatest common divisor's the key
One is what we need to see
For the rule to guarantee
[Outro]
Equivalence relation strong
Arithmetic properties long
Modular math will lead the way
In foundations every day