SOLUTION: If PPQ (3 digit number) * Q = RQ5Q. Then what are P, Q, R

Multiplication of a 3-digit number by a single digit can be 
performed like these two examples below:

  943                    458
    7                      9
   21 <--7x3              72  <--9x8
  28  <--7x4             45   <--9x5
 63   <--7x9            36    <--9x4
 6601                   4122

So let QxQ=AB and QxP=CD

  PPQ
    Q 
   AB <--QxQ
  CD  <--QxP
 CD   <--QxP
 RQ5Q

B must = Q, since we bring it down to the bottom line

  PPQ
    Q 
   AQ <--QxQ
  CD  <--QxP
 CD   <--QxP
 RQ5Q 

We need Q so that QxQ ends in Q
1x1=1, 2x2=4, 3x3=9, 4x4=16, 5x5=25, 6x6=36, 7x7=49, 8x8=64, 9x9=81

Since QxQ must end in Q, Q is 6, since Q obviously can't be 1.
Therefore AQ must be 36.

  PP6
    6 
   36 <--6x6
  CD  <--6xP
 CD   <--6xP
 R656

In order to get the 5 on the bottom line, D must be 2.

  PP6
    6 
   36 <--6x6
  C2  <--6xP
 C2   <--6xP
 R656

In order to get the left-most 6 on the bottom line, C must be 4,

  PP6
    6 
   36 <--6x6
  42  <--6xP
 42   <--6xP
 R656
 
and then P must be 7, since 6x7=42

  776
    6 
   36 <--6x6
  42  <--6x7
 42   <--6x7
 R656

Finally R can only be 4

  776
    6 
   36 <--6x6
  42  <--6x7
 42   <--6x7
 4656

Edwin