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Number Theory

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Moderate

The Fundamental Theorem of Arithmetic, $\gcd$ and $\text{lcm}$

NUMTH-H2UOF0

Let $a, b \in \mathbb{N} \backslash \{1\}$. It follows from the Fundamental Theorem of Arithmetic that $a = p_{1}^{\alpha_1} \cdots p_{s}^{\alpha_{s}}$ and $b = p_{1}^{\beta_1} \cdots p_{s}^{\beta_{s}}$ for some $s \in \mathbb{N}$, where $p_1, \cdots, p_{s}$ are primes and $\alpha_1, \cdots, \alpha_{s}, \beta_1, \cdots, \beta_{s} \in \mathbb{N} \cup \{0\}$. Which of the following statements are true? Select ALL that apply.

A

$\gcd(a, b) = p_{1}^{\min(\alpha_1, \beta_1)} \cdots p_{s}^{\min(\alpha_{s}, \beta_{s})}$, where $\gcd(a, b)$ is the greatest common divisor of $a$ and $b$ and $\min(c, d)$ is the minimum of $c$ and $d$

B

$\text{lcm}(a, b) = p_{1}^{\max(\alpha_1, \beta_1)} \cdots p_{s}^{\max(\alpha_{s}, \beta_{s})}$, where $\text{lcm}(a, b)$ is the least common multiple of $a$ and $b$ and $\max(c, d)$ is the maximum of $c$ and $d$

C

$ab = \gcd(a, b)~\text{lcm}(a, b)$, where $\gcd(a, b)$ is the greatest common divisor of $a$ and $b$ and $\text{lcm}(a, b)$ is the least common multiple of $a$ and $b$

D

$\gcd(a, b) = p_{1}^{\max(\alpha_1, \beta_1)} \cdots p_{s}^{\max(\alpha_{s}, \beta_{s})}$, where $\gcd(a, b)$ is the greatest common divisor of $a$ and $b$ and $\max(c, d)$ is the maximum of $c$ and $d$

E

$\text{lcm}(a, b) = p_{1}^{\min(\alpha_1, \beta_1)} \cdots p_{s}^{\min(\alpha_{s}, \beta_{s})}$, where $\text{lcm}(a, b)$ is the least common multiple of $a$ and $b$ and $\min(c, d)$ is the minimum of $c$ and $d$