Find a complete set of mutually incogruent solutions. 9x≡12±od{15}

Question

Find a complete set of mutually incogruent solutions.  9x≡12±od{15}

lobeflepnoumni · Accepted Answer

Step 1
Since 15 is not prime number .Prime factorization of 15 is

5 \times 3

. Convert the given equation in

b mod 5

and mod 3.
The solution of the equation

9 x \equiv 12 \pm o d 3

and

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

9 x \equiv 12 \pm o d 3

.
The equivalent meaning of above equation is

\frac{3}{9 x} - 12

that is

\frac{3}{3} (3 x - 4)

. Since 3 divide 9x−12 always inn-respective of values of

x \in Z

. Therefore solution of equation is

Z

.
Now solve the equation

9 x \equiv 12 \pm o d 5

.

9 x \equiv 12 \pm o d 5

4 x \equiv 2 \pm o d 5

- x \equiv 2 \pm o d 5

x \equiv - 2 \pm o d 5

x \equiv 3 \pm o d 5

The solution of the equation

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

\leq f t {5 k + 3 . k \in Z r i g h t}

.

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