# Thread: Proof using Eulers Theorem

1. ## Proof using Eulers Theorem

Prove that (2^15)-(2^3) divides (a^15)-(a^3)?

I know that (2^15)-(2^3)=5*7*8*9*13

2. Originally Posted by MichaelG
Prove that (2^15)-(2^3) divides (a^15)-(a^3)?

I know that (2^15)-(2^3)=5*7*8*9*13

Hint: $a^{15}-a^3=a^3\left(a^{12}-1\right)=a^3(a^6-1)(a^6+1)=a^3(a^3-1)(a^3+1)(a^6+1)=$ $a^3(a-1)(a^2+a+1)(a^3+1)(a^6+1)$ ...

Well, now just prove that the above is divisible by 5,7,8,9 and 13 no matter what a is.

Tonio