?

Abstract Algebra

Free Version

Upgrade subject to access all content

Difficult

Primes, Powers, and Factorials

ABSALG-3L4SVH

Which of the following statements are TRUE? You do not need a calculator.

You may use the fact that

$461$ is a prime number

You may also use Wilson's Theorem:

$(p-1)!:=\prod_{r=1}^pr\equiv\,-1$ mod $p$, for $p\ge 2$ prime

Select ALL that apply.

A

We have $7^{461}+8^{461}\equiv 15^{461}$ mod $461$.

B

The ordered list :

$-1, -2, -3, \ldots, -230$

...is congruent mod $461$, respective entry by respective entry, to the ordered list:

$460, 459, \ldots, 231$

C

$(460)!\not\equiv -1$ mod $461$.

D

$(460)!\equiv (230!)^2$ mod $461$.

E

$(230!)^2\not\equiv -1$ mod $461$.