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Abstract Algebra

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Moderate

Factoring Integers into Products of Prime Numbers

ABSALG-VEXGYV

An integer $p\in{\mathbb Z}$ is prime if $p\not=0, 1, -1$ and whenever $p$ divides the product $ab$ of two integers $a$ and $b$, then $p$ divides $a$ or $p$ divides $b$.

Which of the following statements is correct?

A

Every integer $n$ can be expressed uniquely as a product of primes.

B

If $n=p_1p_2p_3=q_1q_2q_3$ then $p_1=\pm q_1$, $p_2=\pm q_2$, $p_3=\pm q_3$.

C

If $n=p_1p_2p_3=q_1q_2q_3$ where $|p_1|<|p_2|<|p_3|$ and $|q_1|<|q_2|<|q_3|$ are all primes,
then $p_1=\pm q_1$, $p_2=\pm q_2$, $p_3=\pm q_3$.

D

If $n=p_1p_2=q_1q_2q_3$ where $|p_1|<|p_2|$ and $|q_1|<|q_2|<|q_3|$ are all primes, then $p_1=\pm q_1$, $p_2=\pm q_2$.

E

If $n=p_1^3p_2^3=q_1^3q_2^3$, where $|p_1|<|p_2|$ and $|q_1|<|q_2|$ are primes, then $p_i=q_i$, $i=1,2,3$.