Digit Reversal

Submitted by Andrew and Russell Scheidegger, 12/03/1996. This article by Valerio De Angelis and Allen Stenger.

Even in fifth grade one finds questions that please the MathNerds...

Find a five digit number ABCDE such that 4 * (ABCDE) = EDCBA. The digits must be distinct, and A must not be 0.

Hint 1

This kind of problem (called a cryptarithm) is solved by trial and error. Because there are 90,000 five-digit numbers, we could just try them all, but that would take a long time unless we hired a computer to help. The challenge in this kind of problem is to think up clever ways to cut down the number of trials. What are some properties of the numbers ABCDE and EDCBA that you can figure out from the equation?

Hint 2

We know from the equation that EDCBA is a multiple of 4, so it is even, and A must be even. We also know that ABCDE has 5 digits but cannot be an extremely large 5-digit number because 4 times it is still a 5-digit number; for example, it could not be as big as 90,000 because 4 times that is 360,000, a 6-digit number. Use these ideas to limit the possible values for A and E.

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