The given equation is:

$ABC \times DEED = ABCABC$

We are told that ABCABC can be written as $ABC \times 1001$, where ABC is a 3-digit number, and DEED represents a 4-digit number.

The hint given is that ABCABC is divisible by 1001, as 1001 is the product of 13, 11, and 7. So, we can rewrite the equation as:

$ABC \times 1001 = ABCABC$

Now, we know that the number ABCABC is essentially the number ABC repeated twice. Let’s analyze the multiplication:

$ABC \times 1001 = ABCABC$

Since $1001 = DEED$, we have:

$DEED = 1001$

The number 1001 is written as DEED, so $D = 1$ and $E = 0$.

Thus, the correct values are:

$D = 1$, $E = 0$

So, the correct answer is C. $D = 1$, $E = 0$.