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 $N$? 
 A: Form three arithmetic sequences:
1.- Numbers that are divisible by $\,4\,$ and less or equal than $\,400\,$:
$$4,8,12,...\Longrightarrow a_1=4\,\,,\,d=4\Longrightarrow a_n=a_1+(n-1)d=4+(n-1)4\leq 400\Longrightarrow n\leq 100$$
2.- Numbers that are divisible by $\,6\,$ and less or equal than $\,400\,$:
$$6,12,18,...\Longrightarrow a_1=6\,\,,\,d=6\Longrightarrow a_n=6+(n-1)6\leq 400\Longrightarrow n\leq66$$
3.- Numbers that are divisible by $\,4\,$ and $\,6\,$ and less or equal than $\,400\,$:
$$12,24,36,...\Longrightarrow a_1=12\,\,,\,d=12\Longrightarrow a_n=12+(n-1)12\leq 400\Longrightarrow n\leq 33$$
Well, take it now from here: how many positive integers less than $\,400\,$ are not divisible by $4\,,\,6\,$?
A: One approach:
Count the number of numbers divisible by $4 ([\frac N4] )$, then count the number of numbers divisible by $6(\frac N6] )$, Now during these calculations we've calculated nos divisible by $12 ([\frac N{12} ] )$ twice, so we subtract it thereby giving the number of integers divisible by either $4$ or $6$ as $[ \frac N4 ] + [ \frac N6 ] - [\frac N{12}]$. Now subtract this number from $N$ to get required answer.
Note:


*

*I've used Principle of inclusion exclusion.

*$[ \cdot ]$ denote floor function.
A: 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.
