Solution of Linear Congruence/Examples/12 x = 9 mod 6

Example of Solution of Linear Congruence

Let $12 x = 9 \pmod 6$.

Then $x$ has no solution in $\Z$.

Proof

We have that:

\(\ds 12 x\)

\(=\)

\(\ds 9\)

\(\ds \pmod 6\)

\(\ds \leadsto \ \ \)

\(\ds 12 x - 9\)

\(=\)

\(\ds 6 k\)

for some $k \in \Z$

\(\ds \leadsto \ \ \)

\(\ds 12 x - 6 k\)

\(=\)

\(\ds 9\)

Then we have that:

$\gcd \set {12, -6} = 6$

which is not a divisor of $9$.

So, from Solution of Linear Diophantine Equation, no solution exists.

$\blacksquare$

Sources

• 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {4-1}$ Basic Properties of Congruences: Exercise $2 \ \text{(c)}$