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## Alphabet Reversed

ABCDE x 4 = EDCBA

Each of the five letters (A, B, C, D, E) stands for a different digit (0-9).

Find the values of A, B, C, D, and E.

NOTE: Follow-up questions after this problem has been solved:

• Is the solution unique?
• Are there solutions for other "reversal" variations: A x 4 = A or AB x 4 = BA or ABC x 4 = CBA or ABCD x 4 = DCBA or ABCDEF x 4 = FEDCBA, etc.
• In the original problem, can you find solutions if you change the 4 to any of the other digits 0-9?

Hint: Focus on the ends. That is, the product E x 4 ends in A (means A is even?) and the product A x 4 does not involve a carry. Thus, what can you conclude about the values of A and E?

Solution Commentary: Because there is no carry for A x 4, we know A = 2, 1, or 0. Now,A=0 implies E= 1, 2, or 3 (due to carry from B x 4). But, A=0 implies E=5 since E x 4 ending in 0. Thus, A must be 1 or 2. But, A must be even by E x 4, which implies A=2 and E is either 3 or 8. But E=3 cannot work with A x 4 where A=2. Thus, E=8.

We note that there is no carry from B x 4, which implies B = 0 or 1. But, B must equal 1 as (D x 4) + 3-carry is odd...which implies D = 7.

Finally, a carry of 3 is necessary to make D=7 for B x 4, which implies C=9.

Check: Does 21978 x 4 = 87912?
Is it the only solution...i.e. does the above logic "prove" no other possibilities will work?

Now, go on to explore the other questions asked...