Question

In the multiplication shown below, the digits have been replaced by
letters: different letters represent different digits, and the
same letters represent the same digits. What was the origianl
multiplication?

ABCDE
x 4
======
EDCBA

please help tried it many times can't get it. It is drivin me nuts.

Answer 1

Someone recently asked and someone else answered the similar question
ABCD x 4 = DCBA

In that case there was only one answer. It took quite a bit of reasoning to come up with the answer, 2178 x 4 = 8712

In your case, it is likely that there is no answer. The longer the string of numbers, the less likely it is that there is an answer. The problem has to be done by a "cut and try" method. If you try to do it with algebra, you end up with one equation and 4 unknowns.

Answer 2

ABCDE x 4 = EDCBA

21978 x 4 = 87912

You just use the fact that the last digit of a multiple of 4 is even. It is not zero, because A = 0 would mean that the number ABCDE is actually a four digit number. It cannot exceed two otherwise you get a six digit number when multiplying it by four. So:

A = 2

This means that the number will be EDCBA will be larger than 80,000 so E has to be 8 or 9. But E = 9 as the first digit of ABCDE
yields A = 6, so E must be 8. You then simply work your way through the multiplication. It is straightforward to come up with the answer (i.e., if it exists as in this case).