Given any $n$ integers $m_1,m_2,\ldots m_n\in \mathbb{N}$ if we define an equivalence relation $\sim$ on $\mathbb{N}^n$ as follows:
$$(a_1,a_2,\ldots a_n)\sim (b_1,b_2,\ldots b_n)\iff \text{For every integer }1\leq k\leq n\text{ we have }a_k\equiv b_k\bmod m_k$$
Then what is the cardinality of the quotient set $\mathbb{N}^n/\sim$?
For the special case when $m_1,m_2,\ldots m_n$ are pairwise coprime, I know $|\mathbb{N}^n/\sim|=m_1m_2\cdots m_n$. However what about in general when $m_1,m_2,\ldots m_n$ might have common factors?
The answer is unconditionally $m_1...m_n$. You did not place any restrictions between the entries of the $n$-tuple.