SOLUTION: If You Multiply 4321 By A Certain 4-digit Number, The 8-digit Answer You Get Ends In Its Reverse

In the multiplication below, the "0"'s
are zeros, and the letter "O" is not
used, and the small a,b,c are not the
same as the capital letters A,B,C.

      4321
      ABCD
     EFGHI
    JKLMN0
   PQRST00
  UBWXY000
  Zabc1234

We see that I=4, thus D=4 and
EFGHI = 17284

 
      4321
      ABC4
     17284
    JKLMN0
   PQRST00
  UBWXY000
  Zabc1234

We see that N=C and N=5 so that
8+5=13, so we put 5's for C and N,
and remember that 1 is carried to
the third column in the addition: 

      4321
      AB54
     17284
    JKLM50
   PQRST00
  UBWXY000
  Zabc1234

Now we see that JKLM = 2160
    
      4321
      AB54
     17284
    216050
   PQRST00
  UBWXY000
  Zabc1234

Now since there was a 1 carried
to the third column, T=9, and that
makes B=9 as well, and we remember
that a 1 is carried to the 4th
column in the addition:

      4321
      A954
     17284
    216050
   PQRS900
  UBWXY000
  Zabc1234

Now we see that PQRS = 3888

      4321
      A954
     17284
    216050
   3888900
  UBWXY000
  Zabc1234

If we remember that ther was a 1
carried to the 4th column then when
we add 1+7+6+8+Y to get a number
ending in 1, Y must be 9 so that
1+7+6+8+9=31, with 3 to carry to the
5th column.  And so A=9

      4321
      9954
     17284
    216050
   3888900
  UBWX9000
  Zabc1234

Now we have the answer, 9954, but
here are the rest of the digits:

      4321
      9954
     17284
    216050
   3888900
  38889900
  43011234

Edwin