This is bit different for other motivation related questions, so I am defining it clearly. First of all,
I am asking motivation for starting to do (elementary/beginning) number theory - which includes very basic topics like congruences, divisibility, linear diophantine equations and theorems of Fermat and Wilson.
But, here's what I mean by motivation:
Give me an example of curious and semi-concrete stuff, which mechanism is buried deep inside those mathematics.
For example:
As a motivation for learning ordinals and set theory, I found working on the hydra game (that you can always kill the hydra !) a very useful motivation. For doing group theory, doing tricks on Rubik's cube was a very interesting motivation.