How many positive integers less than $N$ are not divisible by $4$ or $6$ for some $N$?
[Math] Positive integers less than $N$ not divisible by $4$ or $6$
combinatorics
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Best Answer
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 $4\,,\,6\,$?