Question

If ABC X DEED = ABC ABC, where A, B, C, D, and E are different digits. The values of D and E are:

• A. D = 2, E = 0
• B. D = 0, E = 1
• C. D = 1, E = 0
• D. D = 1, E = 2

Hint:

When solving such puzzles, trial and error can often lead to the correct solution.

Answer 1

The Correct Option is C

To solve the problem, we need to find the different digits A, B, C, D, and E such that the equation ABC × DEED = ABCABC holds. Furthermore, all the digits must be unique. Given that ABCABC is a six-digit number comprised of the repeated three-digit number ABC, we know that the repeating pattern indicates ABCABC can be expressed as 1001 × ABC. Consequently, the equation becomes

ABC × DEED = 1001 × ABC

Now, we can simplify this by dividing both sides by ABC, assuming ABC ≠ 0:

DEED = 1001

The number DEED is composed of the digits D and E, repeated as such: DEED = D × 1000 + E × 100 + E × 10 + D
=> DEED = 1000D + 110E + D
=> DEED = 1001D + 110E
Since the value of DEED is given as 1001, we can equate:

1001D + 110E = 1001

Now, solving for D and E, we first divide both sides by 11 to simplify the equation:
=> (91D + 10E) = 91

This equation can be solved by substituting possible values for D and E. Since they must be different digits, we trial the options provided:

• If D = 1, E = 0: 
  => 91(1) + 10(0) = 91, which satisfies the equation.
• For other options:
  • D = 2, E = 0: 91(2) + 10(0) = 182 ≠ 91
  • D = 0, E = 1: 91(0) + 10(1) = 10 ≠ 91
  • D = 1, E = 2: 91(1) + 10(2) = 111 ≠ 91

Thus, the solution that fits the given equation is D = 1 and E = 0.