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Abstract Algebra

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Prime Numbers: characterizations of positive rational primes

ABSALG-XNMIN9

Which of the following properties characterize positive prime numbers $p$?

A

$p>0$ and $p$ has no positive integer divisors except $1$ and $p$.

B

$p>1$ and if $p\mid ab$ for integers $a$, $b$, then $p\mid a$ or $p\mid b$.

C

$p>1$ and $p$ cannot be factored into a product of two complex numbers.

D

$p>1$ and if $p\mid abc$, for integers $a$, $b$, $c$, then $p\mid a$ or $p\mid b$.

E

$p>2$ and has no positive divisors except $1$ and $p$.