What does A B C D E ‾ \overline{ABCDE} represents?

Number Theory Level 3

In the following addition, different letters represent different non-zero digits. What is the 5-digit number A B C D E ‾ ? \overline{ABCDE}?

A B C D E D B + B C D E D B + C D E D B + D E D B + E D B + D B + B A A A A A A A \begin{array} { l l l l l } & & A & B & C & D & E & D & B \\+ &&&B & C & D & E & D & B \\ + &&&& C & D & E & D & B \\ + &&&&& D & E & D & B \\ + &&&&&& E & D & B\\ + &&&&&&&D & B \\ + &&&&&&&& B \\ \hline & \\ &&\ A & A & A & A & A & A & A \end{array}

What is A B C D E ‾ ? \overline{ABCDE}?

Problem Courtesy Math Meets.


The answer is 84269.

Hana Wehbi by Hana Wehbi , 51, USA

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1 solution

Amos Vagg
Oct 10, 2018

I am only a 13-year-old student so I only have a limited mathematical understanding. Therefore, I solved this by a process of logical elimination, rather than sound mathematical principle. I found that if A=1, B must =3 because there are 7 B's and the number must end in 1 so 7x3=21. In this first round of elimination, all I did was eliminate A=5 as a possibility because all the letters represent different numbers, so the only possible solution of B=5 was actually invalid.

I then proceeded to D, by writing what number would be carried over from B for each question, and discovered that the only possible answers were A=2 B=6 D=8, A=8 B=4 D=1 or A=8 B=4 D=6. My next task was to carry over the remaining numbers to figure what value E could be. I then realised that the only possible answer was A=8 B=4 D=6 because neither 5 could be added to 5 or 0 to result in 2 or 0 be added to 5 or 0 to result in 8.

I then figured out what values of E would be possible, based on the fact I had previously discovered D=6. I knew, therefore, that the total of D in the fourth column had to be 24 and because A=8, we had to carry 4 over from E, meaning the value of E would have to be 9 (9x5=45).

I then used the same elimination to find that the value of C would be 2 and checked my answers by substituting them into the equation.

Therefore A=8 B=4 C=2 D=6 E=9, resulting in 8426964+426964+26964+6964+964+64+4=8888888, giving the final answer of 84269.

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Thank you for sharing your solution.

Hana Wehbi - 2 years, 8 months ago

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