Let the whole number with the digit 4 removed be N.

Then the whole number = 10N+4

4 times this whole number = 4(10N+4) = 40N+16

This must equal to p = N + 4*10^a, where a is some integer. So





The right side is 40000000...0, but we don't know what "a" is so
we don't know where the 0's stop.

However we do know that 

dividend = quotient*divisor + remainder

 4*10^a  =        N*39      +    16

40000... =        N*39      +    16

So we start dividing 39 into 40000000... and if we ever get a remainder
of 16, we will have our answer for N  So let's see if we can get a 16
remainder:
             10256      
         39)400000000000...
            39
             10
             00
             100
              78
              220
              195
               250
               234
                16    <--aha! we got a remainder of 16.

So N, the number without the 4 on the right end is 10256

So M = 102564 and p = 4M = 410256

Edwin