Question

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

Congruence
ANSWERED
asked 2021-03-15
Find a complete set of mutually incogruent solutions.
\(\displaystyle{9}{x}\equiv{12}\pm{o}{d}{\left\lbrace{15}\right\rbrace}\)

Expert Answers (1)

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 \(\displaystyle\le{f}{t}{\left\lbrace{5}{k}+{3}.{k}\in{\mathbb{{{Z}}}}{r}{i}{g}{h}{t}\right\rbrace}\).
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