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.