Correct Option

The given equation is ABC × DEED = ABCABC. To determine the values of D and E, we first analyze the structure of the number ABCABC.

The number ABCABC can be expressed as:

ABCABC = ABC000 + ABC
ABCABC = (ABC × 1000) + (ABC × 1)
ABCABC = ABC × (1000 + 1)
ABCABC = ABC × 1001

Now, substitute this expression back into the original equation:

ABC × DEED = ABC × 1001

Since ABC represents a three-digit number, it is non-zero. Therefore, we can divide both sides of the equation by ABC:

DEED = 1001

By comparing the digits of DEED with 1001, we can deduce the values of D and E:

The first digit D corresponds to 1.
The second digit E corresponds to 0.
The third digit E corresponds to 0.
The fourth digit D corresponds to 1.

Thus, D = 1 and E = 0. This solution is consistent with the condition that A, B, C, D, and E are different digits, as 1 and 0 are distinct, and A, B, C can be other distinct digits (e.g., 2, 3, 4).

Incorrect Options

Options 1 (D = 2, E = 0), 2 (D = 0, E = 1), and 4 (D = 1, E = 2) are incorrect because they do not satisfy the derived identity DEED = 1001. If we substitute the values from these options into the DEED format, the resulting number would not be 1001. For example, if D = 2 and E = 0 (Option 1), DEED would be 2002, which is not equal to 1001. Similarly, the other incorrect options also fail to match the mathematically derived values for D and E.