- #1
Math100
- 802
- 221
- Homework Statement
- Find all the solutions of the congruence ## 7x^2+15x\equiv 4\pmod {111} ##.
- Relevant Equations
- None.
Consider the congruence ## 7x^2+15x\equiv 4\pmod {111} ##.
Note that ## 4\cdot 7=28 ##.
Then ## 7x^2+15x\equiv 4\pmod {111}\implies 196x^2+420x-112\equiv 0\pmod {111} ##.
Observe that
\begin{align*}
&196x^2+420x-112\equiv 0\pmod {111}\\
&\implies (14x)^2+2(14x)(15)+15^2-112-225\equiv 0\pmod {111}\\
&\implies (14x+15)^2\equiv 337\pmod {111}\\
&\implies (14x+15)^2\equiv 4\pmod {111}\\
&\implies (14x+15)^2\equiv 2^{2}\pmod {111}\\
&\implies 14x+15\equiv \pm2\pmod {111}.\\
\end{align*}
Now we consider two cases.
Case #1: Let ## 14x+15\equiv 2\pmod {111} ##.
Then ## 14x\equiv -13\pmod {111} ##.
Thus ## 14x\equiv 98\pmod {111}\implies x\equiv 7\pmod {111} ##.
Case #2: Let ## 14x+15\equiv -2\pmod {111} ##.
Then ## 14x\equiv -17\pmod {111}\implies 14x\equiv 94\pmod {111}\implies 7x\equiv 47\pmod {111} ##.
Thus ## 7x\equiv 602\pmod {111}\implies x\equiv 86\pmod {111} ##.
Therefore, the solutions of the congruence ## 7x^2+15x\equiv 4\pmod {111} ## are ## x\equiv 7\pmod {111} ## and ## x\equiv 86\pmod {111} ##.
Note that ## 4\cdot 7=28 ##.
Then ## 7x^2+15x\equiv 4\pmod {111}\implies 196x^2+420x-112\equiv 0\pmod {111} ##.
Observe that
\begin{align*}
&196x^2+420x-112\equiv 0\pmod {111}\\
&\implies (14x)^2+2(14x)(15)+15^2-112-225\equiv 0\pmod {111}\\
&\implies (14x+15)^2\equiv 337\pmod {111}\\
&\implies (14x+15)^2\equiv 4\pmod {111}\\
&\implies (14x+15)^2\equiv 2^{2}\pmod {111}\\
&\implies 14x+15\equiv \pm2\pmod {111}.\\
\end{align*}
Now we consider two cases.
Case #1: Let ## 14x+15\equiv 2\pmod {111} ##.
Then ## 14x\equiv -13\pmod {111} ##.
Thus ## 14x\equiv 98\pmod {111}\implies x\equiv 7\pmod {111} ##.
Case #2: Let ## 14x+15\equiv -2\pmod {111} ##.
Then ## 14x\equiv -17\pmod {111}\implies 14x\equiv 94\pmod {111}\implies 7x\equiv 47\pmod {111} ##.
Thus ## 7x\equiv 602\pmod {111}\implies x\equiv 86\pmod {111} ##.
Therefore, the solutions of the congruence ## 7x^2+15x\equiv 4\pmod {111} ## are ## x\equiv 7\pmod {111} ## and ## x\equiv 86\pmod {111} ##.