Notation: We say that $a$ divides $b$, and denote it by $a|b$, if there is an integer $k$ such that $b = ka$, for integers $a$ and $b$.
I know that $$a_1|b, \dotsc, a_n|b \implies a_1\cdot\dotsb\cdot a_n|b$$ is not true for every $a_1, \dotsc, a_n$ but what is the additional hypothesis on the $a_i$'s such that the turns it on a valid result, and how to use this hypothesis?
For example, I'm trying to show that (but that's not my question here, i.e., I'm not interested in a solution for this in another way) $42|n^7-n=:d$ for every integer $n$. I already showed that $2|d$, $3|d$ and $7|d$ and I need this result to show that these imply $42|n^7-n$.
I tried to use $\gcd(a_1,\dotsc,a_n)=1$ but I'm not getting anywhere.
I know, also, that $$a \equiv b \pmod{m_1},\dotsc, a \equiv b \pmod{m_k} \implies a \equiv b \pmod{\operatorname{lcm}[m_1,\dotsc,m_k]},$$ but the proof of this uses the fact I'm trying to prove.
If the $a_i$ are pairwise relatively prime, then you get what you want.
This is proved by induction from the theorem that if $a$ and $b$ are relatively prime then $a|c$ and $b|c$ implies that $ab|c$.
This may also be necessary.