Answer:
2730
Step-by-step explanation:
We want to determine the greatest common divisor of the elements of the set [tex]S = \{ n^{13} - n \mid n \in \mathbb{Z} \}.[/tex]
We apply the Fermat's little theorem which states that if p is a prime number, then for any integer a, the number aᵖ − a is an integer multiple of p.
Now, [tex]n^{13} \equiv n \mod p [/tex] if p-1 divides 12.
Since the of 12 are 1,2,3,4, 6, 12, the corresponding primes are 2, 3, 5, 7, 13.
Therefore, the gcd of the elements in [tex]2^{13}-2[/tex] and [tex]3^{13}-3$ is 2 \cdot 3 \cdot 5 \cdot 7 \cdot 13.[/tex]
2*3*5*7*13=2730
Therefore, the gcd of the elements in set S is 2730.
