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https://gateoverflow.in/27341/tifr2014-b-16z
In this question, why ($\mathbb{N},∣)$ is not a complete lattice?

For $any \ finite$ subset of $\mathbb{N}$, $LCM$ of its elements will be $lub$ and $HCF$ will be $glb$ and these $LCM$ and $HCF$ will also be the elements of $\mathbb{N}$.
Even for $any \  infinite$ subset, $HCF$ will be $glb$ and $0$ will be $lub$.

Then why it's not complete lattice?
The only reason I could come up with is they might not considering $0 \ \epsilon \ \mathbb{N}$.
Is there any other reason?

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