Linear congruence $8x\equiv 21\pmod{24}$

Question

$8x\equiv 21\pmod{24}$

I am having problem solving this linear congruence because 8 is even and 21 is odd.

Answer 1

Method 1:

Just do it.

$8x \equiv 21 \pmod {24}$ so there is an integer $k$ so that

$8x = 21 + k24$

$x = \frac {21}8 + 3k$.

But $3k$ is an integer and $\frac {21}8$ is not.

So there is no solution.

Method 2:

We may have trouble as $8,21$ and $24$ aren't coprime so we will use the chinese remainder theorem.

If $a \equiv b \pmod {mn}$ then $a\equiv b \pmod m$ and $a \equiv b\pmod n$ (that's a basic result: If $mn|a-b$ then $m|a-b$ and $n|a-b$).

So we know $8x \equiv 21 \pmod 3$

$2x \equiv 0 \pmod 3$ and $x\equiv 0 \pmod 3$.

And we know $8x \equiv 21 \pmod 8$ so $0\equiv 5\pmod 8$ and that just isn't true. There is no solution.

Method 3:

By Bezout we know that $ax + by = c$ will have solutions if and only if $c$ is a multiple of $\gcd(a,b)$.

Another way of putting that is $ax \equiv c\pmod b$ if and only if $c$ is a multiple of $\gcd(a,b)$.

So $8x \equiv 21 \pmod{24}$ will have solutions if and only if $21$ is a multiple of $\gcd(8,24) = 8$. It is not.

....

Conclusion:

If $\gcd (a,n) = 1$ then there is a unique (up to congruence $\mod n$) $a^{-1}$ so that $a^{-1}\cdot a \equiv 1 \pmod n$.

Therefore $ax \equiv b \pmod n$ will have a unique (up to congruence $\mod n$) solution $x \equiv a^{-1}b \pmod n$.

If $\gcd(a,n) = d\ne 1$ then and $d|b$ then $ax \equiv b \pmod n$ will have $d$ unique (up to congruence $\mod n$) solutions. If you let $a=a'd$ and $n = n'd$ and $b=b'd$ the $a'x \equiv b' \pmod {n'}$ will have a unique solution. Call it $w$. Then $w + n'k; k=0,.....,(d-1)$ will be the $d$ unique solutions to $ax \equiv b \pmod n$.

I'll leave the conclusion's proof to you as an exercise.

Answer 2

There is no solution. In general, the congruence $ax \equiv b \pmod m$ has an integer solution if and only if $b$ is an integer multiple of $\gcd(a,m)$ (for integer $a$ and integer $m \ge 1$).