Finding all solutions for to the equation $x^3 = 0\ {\rm mod}\ 9$

Question

How do I go about finding the solutions to:

$$ x^3 = 0\mod 9 $$

Any help is greatly appreciated thank you

Answer 1

$0^3 = 0[9]$

$1^3 = 1[9]$

$2^3 = 8[9]$

$3^3 = 0[9]$

$4^3 = 1[9]$

$5^3 = 8[9]$

$6^3 = 0[9]$

$7^3 = 1[9]$

$8^3 = 8[9]$

so your solution will be $x = 3n$ such that $n= 0, 1, 2, ...$

Answer 2

Hint: write $x = 3^p y$ with $3\nmid y$.

Answer 3

$9$ divided $x^3$, thus $3$ divides $x^3$. By Euclid's lemma we find that $3$ divides $x$, thus we can define $x=3t$ and this is our solution.