If $x\equiv0\pmod a$ and $x\equiv k\pmod b$, is there a simple expression for $x \pmod {ab}$?

Asked Mar 28 '24 at 20:06

Active Mar 28 '24 at 20:07

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Suppose I know that for some values $x,a,b$ that $$ x \equiv 0\pmod a $$ $$ x \equiv k\pmod b. $$

Is there a simple expression that I can use to get the value of $x$ $\pmod{ab}$ assuming that $a,b$ are coprime?

I know from prior threads that I can use the extended euclidean algorithm to find this value, however I wanted to know if the additional information that $x=0\mod a$ can make it so we can get a solution without having to run the algorithm and rather just evaluate some expression

edited Mar 28 '24 at 20:07

Thomas Andrews

177,126

asked Mar 28 '24 at 20:06

wjmccann

3,085

There might be no solutions or multiple solutions if $a,b$ are not relatively prime. – Thomas Andrews Mar 28 '24 at 20:10
So, you know that the answer is $a\times n$ for some $n\in {0, 1, \cdots, (b-1)}$, but all that tells you is that you need to solve $n\times a \equiv k\pmod b$, so you are still stick inverting $a\pmod b$. – lulu Mar 28 '24 at 20:11
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If you have $\gcd(a,b)=1,$ thenyou can solve $au\equiv 1\pmod b$ and get $x\equiv kau\pmod {ab}.$ – Thomas Andrews Mar 28 '24 at 20:12
In any event, there is an algorithm for solving these sorts of problem, when they are relatively prime, high is covered as the Chinese Remainder Theorem. – Thomas Andrews Mar 28 '24 at 20:46
By the linked Easy CRT formula $, x\equiv a(k/a! \bmod b),\pmod{!ab}\ \ $ – Bill Dubuque Mar 29 '24 at 00:49

If $x\equiv0\pmod a$ and $x\equiv k\pmod b$, is there a simple expression for $x \pmod {ab}$?

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