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emmasim [6.3K]
3 years ago
7

Determine the greatest common divisor of the elements of the set \[ s = \{ n^{13} - n \mid n \in \mathbb{z} \}. \]

Mathematics
1 answer:
Kay [80]3 years ago
5 0

Answer:

2730

Step-by-step explanation:

We want to determine the greatest common divisor of the elements of the set  S = \{ n^{13} - n \mid n \in \mathbb{Z} \}.

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, n^{13} \equiv n \mod p 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 2^{13}-2 and 3^{13}-3$ is 2 \cdot 3 \cdot 5 \cdot 7 \cdot 13.

2*3*5*7*13=2730

Therefore, the gcd of the elements in set S is 2730.

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3 years ago
Part 1. Please show work. Please assist me with these math problems. ​
Vitek1552 [10]

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              2) 2,490,924

              3) Neither

<u>Step-by-step explanation:</u>

S_n=S_{n-2}\cdot (S_{n-1}-1)\quad \\\\S_1=2\quad given\\S_2=3\quad given\\S_3=S_1(S_2-1)\quad =2(3-1)\quad =4\\S_4=S_2(S_3-1)\quad =3(4-1)\quad =9\\S_5=S_3(S_4-1)\quad =4(9-1)\quad =32\\S_6=S_4(S_5-1)\quad =9(32-1)\quad =279\\S_7=S_5(S_6-1)\quad =32(279-1)\quad =8,896\\S_8=S_6(S_7-1)\quad =279(8896-1)\quad =2,481,705\\S_9=S_7(S_8-1)\quad =8896(2481705-1)\quad =22,077,238,784

\sum^8_{k=1}S_k\\\\=S_1+S_2+S_3+S_4+S_5+S_6+S_7+S_8\\\\= 2,490,924\\

Neither

Not arithmetic because the difference between each of the terms is not the same.            

                        S₂ - S₁          S₃ - S₂           S₄ - S₃      

                      3 - 2 = 1       4 - 3 = 1          9 - 4 = 5

Not geometric because the ratios between each of the terms is not the same.                

                       \dfrac{S_2}{S_1}=\dfrac{S_3}{S_2}\qquad \rightarrow \dfrac{3}{2}=\dfrac{4}{3}\qquad \rightarrow \text{cross multiply to get}: 9\neq 8

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D. 9/DF=18/16.......the answer

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