[Verse 1]
When we say "a divides b" in math
There's a special meaning on this path
It means there exists some integer k
Where b equals a times k, that's the way
If you can multiply a by some whole number true
And get exactly b, then a divides right through
No remainder left, it goes in clean
The most fundamental fact you've ever seen
[Chorus]
A divides b means there's a k
B equals a times k, remember this way
If zero's involved, then a divides it clean
'Cause zero equals a times zero, if you know what I mean
One divides everything, that's always true
Divisibility rules will see you through
[Verse 2]
Let's explore the properties we know
First rule: a divides zero, here we go
Zero equals a times zero every time
So a divides zero, that's our first rhyme
One divides every number that exists
'Cause any number equals one times it, get this?
These basic facts will serve you well
In the divisibility stories we tell
[Chorus]
A divides b means there's a k
B equals a times k, remember this way
If zero's involved, then a divides it clean
'Cause zero equals a times zero, if you know what I mean
One divides everything, that's always true
Divisibility rules will see you through
[Bridge]
Transitivity comes into play
If a divides b and b divides c today
Then a divides c, it follows through
This chain of division will work for you
Linear combinations, here's the key
If a divides b and a divides c, you see
Then a divides b times x plus c times y
For any integers x and y, don't be shy
[Verse 3]
When you see that vertical line so tall
Between two numbers standing proud and tall
Remember the definition crystal clear
There exists an integer k, keep it near
B equals a times k, that's the test
No fractions, no decimals, integers are best
This foundational truth will guide your way
Through number theory every day
[Chorus]
A divides b means there's a k
B equals a times k, remember this way
If zero's involved, then a divides it clean
'Cause zero equals a times zero, if you know what I mean
One divides everything, that's always true
Divisibility rules will see you through
[Outro]
So when you see "a divides b"
Think integer k, and you'll be free
To understand the deeper truth
Divisibility's eternal proof