The missing digit

Complete the missing digit in the number 3 ∗ 43 to form a number divisible by three. If there are multiple options, list them all. (The omitted digit is marked with the symbol ∗. ) Answers must be justified!

Correct answer:

x₁ =  2
x₂ =  5
x₃ =  8

Step-by-step explanation:

$$\begin{gathered} 3+x+4+\frac{3}{3} \\ x+\frac{10}{3} \\ x+\frac{1}{3} \\ {x}_1=2 \end{gathered}$$

$x_2=x_1+3=2+3=5$

$x_3=x_2+3=5+3=8$

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