# Thread: gcd, lcm

1. ## gcd, lcm

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

2. #1: Let $d = (a, a+1) \geq 1$. This means that $d \mid a$ and $d \mid (a+1)$.

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

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

So $d = \cdots$

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