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?

Let A,B,C be the sets of integers between $1$ and $1000$ (inclusive) which are divisible $4,5,6$.  We want $|(A\cup B\cup C)^c|$
Now $$|(A\cup B\cup C)|=|A|+|B|+|C|-|A\cap B|-|B\cap C|-|A\cap C|+|A\cap B\cap C|$$
Now find $$|A|=\lfloor\dfrac{1000}{4}\rfloor=?$$
$$|B|=\lfloor\dfrac{1000}{5}\rfloor=?$$
$$|C|=\lfloor\dfrac{1000}{6}\rfloor=?$$
$$|A\cap B|=\lfloor\dfrac{1000}{20}\rfloor=?$$
and so on .....
