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

ringearV

ringearV

Answered question

2021-01-30

Solve:
x33x2+27 is congruent to 0 (mod 225)

Answer & Explanation

likvau

likvau

Skilled2021-01-31Added 75 answers

Step 1
Consider the following problem x33x2+270(mod225)
Prime factorization of 225 is 32×52
Now, 225=32×52x33x2+27 implies that the polynomial is reducible over F3andF5
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 mod(32×52)
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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