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

ringearV 2021-01-30 Answered
Solve:
${x}^{3}-3{x}^{2}+27$ is congruent to 0 (mod 225)
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## Expert Answer

likvau
Answered 2021-01-31 Author has 75 answers
Step 1
Consider the following problem ${x}^{3}-3{x}^{2}+27\equiv 0\left(\text{mod}225\right)$
Prime factorization of 225 is ${3}^{2}×{5}^{2}$
Now, $225=\frac{{3}^{2}×{5}^{2}}{{x}^{3}-3{x}^{2}+27}$ implies that the polynomial is reducible over ${F}_{3}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{F}_{5}$
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 $\text{mod}\left({3}^{2}×{5}^{2}\right)$
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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