Solving cubic congruence

Question

I wasn't actually taught about cubic congruences equation and was managing with quadratic congruences until I was hit with this: $$x^3 \equiv 53 (\text{ mod } 120)$$ Effort: I tried deconstructing it into a system of congruence equation assuming that there is several congruences towards prime decomposition of $120 = 3\cdot 5\cdot 8$ but ended up using Chinese remainder theorem to solve it into $53$ (which was quite silly).

I also tried putting it into $x^3-53 \equiv 0 (\text{ mod } 120)$ and use Special Algebra Expansions $a^3-b^3$ or $(a-b)^3$ but got stuck. A few useful hints would be appreciated. Thank you.

Answer 1

Using CRT:

• $x^3\equiv_3 53\equiv_3 2\Rightarrow x\equiv_3 2$
• $x^3\equiv_5 53\equiv_5 3\Rightarrow x\equiv_5 2$
• $x^3\equiv_8 53\equiv_8 5\Rightarrow x\equiv_8 5$

This implies that $x\equiv_{15}2$ and $x\equiv_8 5$, which is the same as saying $x-2\equiv_{15}0$ and $x-2\equiv_8 3$. The first number $8k+3$ that is divisible by $15$ is $75$, so $$x\equiv_{120}77$$

Answer 2

$$\text{Just solve } \mkern 100mu\begin{cases}x^3\equiv -1\mod 3,\\x^3\equiv\phantom{-}3\mod 5,\\x^3\equiv-3\mod 8,\end{cases}\qquad\text{then lift the solutions mod. $120$.}$$ This supposes each elementary equation has solutions.