gcd, lcm

**Sea** · Dec 14th 2008, 05:29 PM

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

**o_O** · Dec 14th 2008, 05:49 PM

#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 $[a,b] = ab$

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