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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?

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The answer is unconditionally $m_1...m_n$. You did not place any restrictions between the entries of the $n$-tuple.

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  • $\begingroup$ I don't understand what you mean by restrictions. $\endgroup$ – user3865391 Dec 6 '18 at 20:11
  • $\begingroup$ You say you have an argument when $m_i$ are pairwise coprime. Is there anything that goes wrong with the proof when for instance $n=2$ and $m_1=2,m_2=4$? $\endgroup$ – zoidberg Dec 6 '18 at 20:28
  • $\begingroup$ Please don't downvote posts simply because you don't understand. $\endgroup$ – zoidberg Dec 6 '18 at 20:47

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