Thread: Number theory

Show that if a is an integer, then 3 divides (a to the power 3) - a

I don't know how to use the mathematical software to express the question, I apologize for any inconvenience. Thanks for helping me out.

Originally Posted by suedenation

Show that if a is an integer, then 3 divides (a to the power 3) - a

I don't know how to use the mathematical software to express the question, I apologize for any inconvenience. Thanks for helping me out.

a^3-a=a(a^2-1)=a(a-1)(a+1)

that is it is the product of three consecutive integers. But in every set
of three consectutive integers at least one is divisible by 3.

Hence a^3-a is divisible by 3.

RonL

Furthermore, this is a faboulus beautiful elegant powerful super and amazing theorem of Fermat (my favorite mathemation).

That,
a^p-a
Is always divisible by p