If $ABC\times DEED=ABCABC$; where A, B, C, D and E are different digits, what are the values of D and E?

The correct option is D = 1, E = 0.

Let us analyze the given alphametic/multiplication equation step-by-step to find the values of $D$ and $E$:

$ABC\times DEED=ABCABC$

First, we can express the six-digit number $ABCABC$ in terms of the three-digit number $ABC$. Note that:

$ABCABC=ABC\times 1000+ABC$

By factoring out the common term $ABC$ from the right-hand side of the equation, we get:

$ABCABC=ABC\times (1000+1)$

$ABCABC=ABC\times 1001$

Now, let us substitute this back into our original equation:

$ABC\times DEED=ABC\times 1001$

Since $A$, $B$, and $C$ represent different digits, the three-digit number $ABC$ is non-zero. Therefore, we can divide both sides of the equation by $ABC$:

$DEED=1001$

By comparing the digits of the four-digit numbers on both sides of $DEED=1001$, we find:

The first and last digit, $D=1$.
The two middle digits, $E=0$.

Thus, the values of $D$ and $E$ are $D=1$ and $E=0$.