Bài giải :

Ta có  $\overline{20a20a20a}\,=\overline{20a}.1000000+\overline{20a}.1000+\overline{20a}=\overline{20a}.1001001$

Để $\overline{20a20a20a}\,\,\vdots \,\,7$ thì $\overline{20a}\,\,\vdots \,\,7$  (vì $1001001\,\,\cancel{\vdots }\,\,7$)

Để $\overline{20a}\,\,\vdots \,\,7$ thì $\left( 200+a\, \right)\vdots \,\,7$$\Rightarrow \left( 196+4+a \right)\,\,\vdots \,\,7\Rightarrow \left( 4+a \right)\,\,\vdots \,\,7$ vì  $196\,\,\vdots \,\,7$

Suy ra a = 3.