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Number Theory

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Moderate

Statements Involving $\text{lcm}$ and $\gcd$

NUMTH-HWNDTW

Let $a, b \in \mathbb{N}$.

Which of the following statements about $\gcd(a, b)$ and $\text{lcm}(a, b)$ are true (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 $b$)?

Select ALL that apply.

A

$\text{lcm}(a, \gcd(a, b)) = a$

B

$\gcd(a, \text{lcm}(a, b)) = a$

C

$\text{lcm}(a,b) \leq \gcd(a, b)$

D

$\gcd(a, b) = \text{lcm}(a, b)$ if and only if $a = b$