BenoguigoliB

2021-03-15

Find a complete set of mutually incogruent solutions.
$9x\equiv 12±od\left\{15\right\}$

lobeflepnoumni

Step 1
Since 15 is not prime number .Prime factorization of 15 is $5×3$. Convert the given equation in $b\text{mod}5$ and mod 3.
The solution of the equation $9x\equiv 12±od3$ and
$9x\equiv 12±od5$
Now solve the above system that will be find out by finding common solution of above equations.
Step 2
Solve the equation $9x\equiv 12±od3$.
The equivalent meaning of above equation is $\frac{3}{9x}-12$ that is $\frac{3}{3}\left(3x-4\right)$. Since 3 divide 9x−12 always inn-respective of values of $x\in \mathbb{Z}$. Therefore solution of equation is $\mathbb{Z}$.
Now solve the equation $9x\equiv 12±od5$.
$9x\equiv 12±od5$
$4x\equiv 2±od5$
$-x\equiv 2±od5$
$x\equiv -2±od5$
$x\equiv 3±od5$
The solution of the equation $x\equiv 3±od5$ is 5k+3 where k is an integer. The common solution of both the equation is 5k+3 where k is an integer.
The set form of the solution is $\le ft\left\{5k+3.k\in \mathbb{Z}right\right\}$.

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