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## Homework Statement

Prove that 2^15-2^3 divides a^15-a^3 for any integer a.

Hint: 2^15-2^3 = 5*7*8*9*13

## Homework Equations

fermats theorem

eulers theorem

## The Attempt at a Solution

I think that the problem is equal to show that 4080 divides any number a^13-a^3, that is

a^15-a^3 = k * 5*7*8*9*13

I was thinking about using Eulers theorem on each 5,7,8,9,13

phi(5)=4

phi(7)=7

phi(8)=4

phi(9)=6

phi(13)=12

But my problem is this. This works for all the prime numbers because then gcd(a^15-a^3, prime)=1

But 8 and 9 we do not have prime numbers, so in order to use Eulers theorem i must show that

gcd(a^15-a^3,8)=1 and gcd(a^15-a^3,9)=1, correct?

How do I do this?

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