<s>[INST] Positive integers less than $N$ not divisible by $4$ or $6$ How many positive integers less than $N$ are not divisible by $4$ or $6$ for some $N$? 
 [/INST] A naive and computational answer: look at congruence modulo 12.


*

*Empty (no natural numbers less than 1) = $0$

*${1}$ = $1$

*${1,2}$ = $2$

*${1, 2, 3}$ = $3$

*${1, 2, 3}$ = $3$

*${1, 2, 3, 5}$ = $4$

*${1, 2, 3, 5}$ = $4$

*${1, 2, 3, 5, 7}$ = $5$

*${1, 2, 3, 5, 7}$ = $5$

*${1, 2, 3, 5, 7, 9}$ = $6$

*${1, 2, 3, 5, 7, 9, 10}$ = $7$

*${1, 2, 3, 5, 7, 9, 10, 11}$ = $8$


From here, observe that any $N$ will be congruent to one of these guys, so you add $N \mod 12$ to 8 times the floor of $N/12$; ie $13 = 12 + 1$ so $n(13) = 8 \times 1 + 0 = 8$ and $n(155) = 8 \times 12 + n(11) = 96 + 7 = 103$.
And then see how you can improve my naive method to get a more "general" answer.
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