The integer ababab (a>0,b>0) is always divisible by 7, without rest.

I tried to prove this by:

Solve[Mod[(a*10^5 + b*10^4 + a*10^3 + b*10^2 + a*10^1 + b), 7] == 0, {a, b}, PositiveIntegers]

I expected a True as result. What do I wrong?

What does the result show?

$\{\{a\to \fbox{$7 c_1\text{ if }(c_2|c_1)\in \mathbb{Z}\land c_1\geq 1\land c_2\geq 0$},b\to \fbox{$7 c_2+1\text{ if }(c_2|c_1)\in \mathbb{Z}\land c_1\geq 1\land c_2\geq 0$}\},...\}$

The integer $ababab$ $(a>0,b>0)$ is always divisible by $7$, without rest.

I tried to prove this by:

Solve[Mod[(a*10^5 + b*10^4 + a*10^3 + b*10^2 + a*10^1 + b), 7] == 0, {a, b}, PositiveIntegers]

I expected a True as result. What do I wrong?

What does the result show?

$\{\{a\to \fbox{$7 c_1\text{ if }(c_2|c_1)\in \mathbb{Z}\land c_1\geq 1\land c_2\geq 0$},b\to \fbox{$7 c_2+1\text{ if }(c_2|c_1)\in \mathbb{Z}\land c_1\geq 1\land c_2\geq 0$}\},...\}$