If $x=3 \mod 11$ and $x=3 \mod 19$ is there a modular formula/rule that can combine both statements together?
If $x=5 \mod 9$ is there a modular formula/rule that can convert $x=5 \mod 9$ into $x=r \mod 11$, where r is some arbitrary integer?
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Have you heard http://mathworld.wolfram.com/ChineseRemainderTheorem.html and answer of your last question is no – lab bhattacharjee Sep 29 '13 at 18:22
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For the first part, yes, this is known as the Chinese Remainder Theorem.
For the second part, there cannot be such a formula: for example, $5 \equiv 5 \pmod 9$ and $5 \equiv 5 \pmod {11}$, while at the same time $14 \equiv 5 \pmod 9$ and $14 \equiv 3 \not\equiv 5 \pmod {11}$. Such a formula would need to map $5$ to both $5$ and $3$, which is not possible.
Rebecca J. Stones
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