Question

# Find a complete set of mutually incogruent solutions. 9x\equiv 12\pmod {15}

Congruence
Find a complete set of mutually incogruent solutions.
$$\displaystyle{9}{x}\equiv{12}\pm{o}{d}{\left\lbrace{15}\right\rbrace}$$

2021-03-17
Step 1
Since 15 is not prime number .Prime factorization of 15 is $$\displaystyle{5}\times{3}$$. Convert the given equation in $$\displaystyle{b}\text{mod}{5}$$ and mod 3.
The solution of the equation $$\displaystyle{9}{x}\equiv{12}\pm{o}{d}{3}$$ and
$$\displaystyle{9}{x}\equiv{12}\pm{o}{d}{5}$$
Now solve the above system that will be find out by finding common solution of above equations.
Step 2
Solve the equation $$\displaystyle{9}{x}\equiv{12}\pm{o}{d}{3}$$.
The equivalent meaning of above equation is $$\displaystyle{\frac{{{3}}}{{{9}{x}}}}-{12}$$ that is $$\displaystyle{\frac{{{3}}}{{{3}}}}{\left({3}{x}-{4}\right)}$$. Since 3 divide 9x−12 always inn-respective of values of $$\displaystyle{x}\in{\mathbb{{{Z}}}}$$. Therefore solution of equation is $$\displaystyle{\mathbb{{{Z}}}}$$.
Now solve the equation $$\displaystyle{9}{x}\equiv{12}\pm{o}{d}{5}$$.
$$\displaystyle{9}{x}\equiv{12}\pm{o}{d}{5}$$
$$\displaystyle{4}{x}\equiv{2}\pm{o}{d}{5}$$
$$\displaystyle-{x}\equiv{2}\pm{o}{d}{5}$$
$$\displaystyle{x}\equiv-{2}\pm{o}{d}{5}$$
$$\displaystyle{x}\equiv{3}\pm{o}{d}{5}$$
The solution of the equation $$\displaystyle{x}\equiv{3}\pm{o}{d}{5}$$ 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 $$\displaystyle\le{f}{t}{\left\lbrace{5}{k}+{3}.{k}\in{\mathbb{{{Z}}}}{r}{i}{g}{h}{t}\right\rbrace}$$.