The sequence
$0 \longrightarrow\mathbb{Z}/(a\vee b \vee c)\mathbb{Z} \underset{\phi_1}{\longrightarrow} \mathbb{Z}/a\mathbb{Z} \bigoplus \mathbb{Z}/b\mathbb{Z} \bigoplus \mathbb{Z}/c\mathbb{Z}\underset{\phi_2}{\longrightarrow} \mathbb{Z}/(a \wedge b)\mathbb{Z} \bigoplus \mathbb{Z}/(a \wedge c) \mathbb{Z} \bigoplus \mathbb{Z}/(b\wedge c)\mathbb{Z} \underset{\phi_3}{\longrightarrow} \mathbb{Z}/(a\wedge b \wedge c)\mathbb{Z}\longrightarrow 0$
given by $\phi_1(x)\equiv (x,x,x),\phi_2(x,y,z)\equiv(x-y,x-z,y-z)$ and $\phi_3(x,y,z)\equiv x-y+z$ is exact.
Here $a\wedge b= \mathop{gcd}(a,b)$ and $a\vee b= \mathop{lcm}(a,b)$
The proof I have is tedious but elementary (first one shows that first coordinates of the element of the kernel can be assumed to be zero, then it boils down to some artihmetic fact).
Is there a more general way to view this so as to extend it to more variables ?
As a side note one can use the exactness to compute cardinals and get formulas such as $a\vee b \vee c= \dfrac{abc(a\wedge b\wedge c)}{(a\wedge b)(a\wedge c)(b\wedge c)}$.