How many integers from $1$ through $1000$ are not divisible by any one of $4, 5$, and $6$? I hate to use this resource as an argument settler but my friend and I have come across this question and we cannot agree on an answer. I got $500$ through the use of inclusion and exclusion and he got $466$ through the use of gathering LCM's and such. Who is correct?

Inclusion Exclusion makes more sense to me than "the use of gathering LCM's and such"

But you do need LCM to do inclusion exclusion.

There are $1000$ integers. $250$ divisible by $4$ and $200$ by $5$ and $166$ by $6$. There are $1000/20 = 50$ there are divisible by both $4$ and $5$ (divisible by $4$ and $5$ means divisible by $20$). But to be divisible by both $4$ and $6$ means to be divisible by $12$, not $24$ so there are $83$ divisible by both $4$ and $6$. And $33$ by both $6$ and $5$. And to be divisible by all $3$ is to be divisible by $60$ and there are $16$ of those.

So the answer is $1000 - 250 - 200 - 166 + 50 + 83 + 33 - 16 = 534$.