Thread: gcd, lcm

$\displaystyle a\in\mathbb{Z^{+}}$ (a,a+1)=? , [a,a+1]=? (proof)

#1: Let $\displaystyle d = (a, a+1) \geq 1$. This means that $\displaystyle d \mid a$ and $\displaystyle d \mid (a+1)$.

Fact: If $\displaystyle d \mid x$ and $\displaystyle d \mid y$, then $\displaystyle d \mid (x+y)$.

This means that $\displaystyle d \mid \left[(a+1) - a\right] \ \Leftrightarrow \ d \mid 1$.

So $\displaystyle d = \cdots$

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#2: Use the fact that if $\displaystyle (a,b) = 1$ then $\displaystyle [a,b] = ab$