Shifting the numbers

We can represent the numbers $\overline{ABCDEF}$ and $\overline{CDEFAB}$ by using numbers x and y.

Let's define the numbers x and y as following: $x = \overline{AB}$ and $y = \overline{CDEF}$ .

Now, $\overline{ABCDEF} = 10000x + y$ and $\overline{CDEFAB} = 100y + x$ .

Since $\frac{\overline{ABCDEF}}{\overline{CDEFAB}} = \frac{12}{17}$ is true, we know that $\frac{10000x + y}{100y + x} = \frac{12}{17}$ is also true.

Therefore, $12\left(10000x + y\right) = 17\left(100y + x\right) \Rightarrow 169988x = 1183y \Rightarrow 1868x = 13y$ .

$\overline{ABCDEF}$ reaches it's largest value when $\overline{AB}$ is as large as possible.

We know that x must be a 2-digit number divisible by 13, and y must be a 4-digit number.

Since y must be a 4-digit number $y < 10000 \Rightarrow 1868x < 130000 \Rightarrow x \leq 69$

Since $69 \equiv 4$ (mod 13), and the largest value of $\overline{ABCDEF}$ is achieved when $\overline{AB} = x$ is as large as possible.

Therefore the largest possible value of x is 65, and therefore y is 9340, and therefore $\overline{ABCDEF} = \boxed{659340}$ . $\square$