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Number Theory

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Roots of a Factorization of a Positive Integer Greater than $1$

NUMTH-ZTSRJK

Let $n \in \mathbb{N} \backslash \{1\}$. Then, by the Fundamental Theorem of Arithmetic, $n = p_{1}^{\alpha_1} \cdots p_{s}^{\alpha_{s}}$, for some $s \in \mathbb{N}$ where $p_{1}, \cdots, p_{s}$ are prime and $\alpha_1, \cdots, \alpha_{s} \in \mathbb{N} \cup \{ 0\}$. Which of the following statements are true concerning $n^{1/m}$ where $m \in \mathbb{Z} \backslash \{ 0\}$.

A

$n^{1/m} \in \mathbb{Q}$ if $\frac{\alpha_1}{m}, \cdots, \frac{\alpha_{s}}{m} \in \mathbb{Z}$

B

$n^{1/m} \in \mathbb{N}$ if $\frac{\alpha_1}{m}, \cdots, \frac{\alpha_{s}}{m} \in \mathbb{N}$

C

$n^{1/m} \in \mathbb{N}$ if $\frac{\alpha_1}{m}, \cdots, \frac{\alpha_{s}}{m} \in \mathbb{Z}$

D

$n^{1/m} \in \mathbb{Q}$ if $\frac{\alpha_1}{m}, \cdots, \frac{\alpha_{s}}{m} \in \mathbb{Q}$