Question

Given that the number $\overline{35\alpha 64}$ is divisible by 3, where α is a digit, what are the possible values of α?

Answer 1

$$\begin{gathered} \mathrm{It}\mathrm{is}\mathrm{given}\mathrm{that}\overline{35a64}\mathrm{is}\mathrm{a}\mathrm{multiple}\mathrm{of}3. \\ \therefore (3+5+a+6+4)\mathrm{is}\mathrm{a}\mathrm{multiple}\mathrm{of}3. \\ \therefore (a+18)\mathrm{is}\mathrm{a}\mathrm{multiple}\mathrm{of}3. \\ \therefore (a+18)=0,3,6,9,12,15,18,21... \\ \mathrm{But}a\mathrm{is}\mathrm{a}\mathrm{digit}\mathrm{of}\mathrm{number}\overline{35a64}.\mathrm{So},a\mathrm{can}\mathrm{take}\mathrm{value}0,1,2,3,4...9. \\ a+18=18\Rightarrow \mathrm{a}=0 \\ a+18=21\Rightarrow \mathrm{a}=3 \\ a+18=24\Rightarrow \mathrm{a}=6 \\ a+18=27\Rightarrow \mathrm{a}=9 \\ \therefore a=0,3,6,9 \end{gathered}$$