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# Solve: x^3-3x^2+27 is congruent to 0 (mod 225)

Question
Polynomial factorization
asked 2021-01-30
Solve:
$$\displaystyle{x}^{{3}}-{3}{x}^{{2}}+{27}$$ is congruent to 0 (mod 225)

## Answers (1)

2021-01-31
Step 1
Consider the following problem $$\displaystyle{x}^{{3}}−{3}{x}^{{2}}+{27}\equiv{0}{\left(\text{mod}{225}\right)}$$
Prime factorization of 225 is $$\displaystyle{3}^{{2}}\times{5}^{{2}}$$
Now, $$\displaystyle{225}=\frac{{{3}^{{2}}\times{5}^{{2}}}}{{{x}^{{3}}-{3}{x}^{{2}}+{27}}}$$ implies that the polynomial is reducible over $$\displaystyle{F}_{{3}}{\quad\text{and}\quad}{F}_{{5}}$$
So, there is a solution under 3 with x=0 and a solution under modulo 5 with x=1
Since the modulus is composite , so we solve the equation with $$\displaystyle\text{mod}{\left({3}^{{2}}\times{5}^{{2}}\right)}$$
Therefore, we have
x=0(mod 3)
x=1(mod 5)
Step 2
Now, using Chinese remainder theorem for
x=0(mod 3)
x=1(mod 5)
we get x=6

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