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If we make a table such that each column contains numbers modulo $m$ and each row containing numbers modulo $n$. Let us denote the element $a_{ij}=x$ and $a_{i'j'}=y$ Where $0\leq i,i'<m$ and $0\leq j,j'<n$. $$x\equiv i \quad(mod \;m)$$ $$x\equiv j \quad(mod \;n)$$ $$y\equiv i' \quad(mod \;m)$$ $$y\equiv j' \quad(mod \;n)$$

This is essentially how I translate a program i wrote to generate those tables and it works fine. (loop over every number, then decide its indices). I am now trying to show that no two numbers in the table are same (which is so if $m,n$ are coprime)

Start by supposing that $x=y$ then, $$i\equiv i' \quad(mod \;m)$$ $$j\equiv j' \quad(mod \;n)$$

But because $0\leq i,i'<m$ and $0\leq j,j'<n$, we have $i=i'$ and $j=j'$. Hence no two numbers in this table are the same.

However, there has to be a mistake here as I never took into account $m$ and $n$ to be coprime.

kuch nahi
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2 Answers2

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The problem when $m$ and $n$ are not coprime is that some positions in the table cannot be filled; for instance, if $m=6$ and $n=10$, $a_{01}$ is undefined. Your argument correctly shows that the entries that can be defined must all be distinct.

Brian M. Scott
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  • But the problem is, for $m=6$ and $n=10$ we have $;a_{00}=0\mbox{ and }30$. So for an entry, the above "proof" of mine can also be reversed to show that if $i=i'$ and $j=j'$ then $x=y$, which is clearly not true as we can have two different numbers in a position. – kuch nahi Jul 16 '11 at 18:11
  • @kuch nani: You are confusing the issue of whether, given one specific way of filling in the table, two entries can be the same, vs. whether there is more than one way of filling in the table. It is true that there is more than one way of filling in the table; but Brian's point is that, given two entries in different locations, they cannot be equal. – Zev Chonoles Jul 16 '11 at 18:17
  • @Kuch: Why do you think that it reverses? $x=y$ implies that $i\equiv i'\pmod m$ and $j\equiv j'\pmod n$, but the reverse implication is false. – Brian M. Scott Jul 16 '11 at 18:18
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It is an immediate consequence of uniqueness of solutions in CRT (Chinese Remainder Theorem). The proof is easy. If $\,\gcd(m,n) = 1\,$ then $\ x\bmod mn\ \to\ (x\bmod m,\ x\bmod n)\ $ is $\,1$ to $1.\ $ For if $\ x,\,y\ $ map to the same value then $\, x\equiv y\pmod{\!m}\, $ so $\ m\ |\ x-y\,,\,$ i.e. $\,m\,$ divides $\,x-y\,.\,$ Similarly $\,n\ |\ x-y\,.\,$ Hence $\, {\rm lcm}(m,n)\mid x-y\,.\,$ But $\,\gcd(m,n) = 1 \!\iff\! {\rm lcm}(m,n) = mn.\,$ Thus $\ mn = {\rm lcm}(m,n)\ |\ x-y\,,\,$ hence $\, x \equiv y\pmod{\!mn}\,.\,$ Hence indeed the map is $1$ to $1$.

Conversely, if $\,{\rm lcm}(m,n) < mn\ $ then $\,x\,$ and $\ x + {\rm lcm}(m,n)\ $ are not congruent $\!\bmod\ mn.\,$ However, both map to the same value $\ (x\bmod m,\, x\bmod n),\,$ since both $\,m,n\mid {\rm lcm}(m,n)\,.\ $ This explains the example you gave in your comment, where $\,m,n = 6,10\,$ so $\,{\rm lcm}(m,n) = 30,\,$ hence $\, x= 0\,$ and $\,x+30 = 30\,$ both map to $\,(0,0)\,$ but $\, 30\not\equiv 0\pmod{\!60}$.

The existence of solutions is also easy. Clearly $\ |\mathbb Z\bmod mn| = mn = |\mathbb Z\bmod m \times \mathbb Z\bmod n|,\, $ thus the above map being $1$ to $1$ on these equal cardinality finite sets implies the map is also onto, i.e. every pair $\,(i\bmod\ m,\,j\bmod n)\,$ is the image of some $\, x\bmod\ mn.\,$

In summary: if $\,\gcd(m,n) = 1$ then the congruence system $\ x\equiv i\pmod{\!m},\ x\equiv j\ \pmod{\!n}\, $ has a unique solution $\!\bmod\ mn,\,$ i.e. if $\,x',x\,$ are solutions then $\,x'\equiv x\pmod{\!mn}.\,$ For a simple constructive proof see here.

Bill Dubuque
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