How would I solve this? I don't understand this question. Or more precisely how they derived the answer for the examples given. Can someone explain? Thanks.
E.g. All integers can be represented using the base B =-10 using the digits 0, 1, 2...9 and without using a negative sign in front of the number.
For example, -1467 = (2673) subscript(-10) and 10 = (190)subscript(-10)
a) what decimal numbers do (56)sub(-10) represent?
b)Determine the base B= -10 representations of the decimal numbers -209?

 A: If you think of regular decimal notation, $456_{10}=4\cdot10^2+5\cdot10^1+6\cdot 10^0$.  In any other base $b$, you do the same thing:  $456_{b}=4\cdot b^2+5\cdot b^1+6\cdot b^0$.  There is no requirement that $b$ be positive, so you can plug in $b=-10$.  Then check that $-1467_{10}=2\cdot (-10)^3+6 \cdot (-10)^2+7 \cdot (-10)^1+3\cdot (-10)^0$  The conversion is done the same way you convert bases as well.  It's fun to play with.
A: If you understand binary or decimal, you should be able to understand this because the principle is the same.
Take for instance 9 in base 2, this is 1001 because $9=1*2^3+0*2^2+0*2^1+1*2^0$.
Likewise, 249 in base 10 is 249 because $249=2*10^2+4*10^1+9*10^0$.
The same logic is true in base -10. -1467 is equal to 2673 because $$-1467=2*(-10)^3+6*(-10)^2+7*(-10)^1+3*(-10)^0$$
A: Since you tagged your question as mathematica, here you have a Mathematica function to solve it:
negBase[base_, number_] := 
 With[{n = Length@IntegerDigits[number, -base] + 1}, 
  Array[x, n, 0] /. Solve[{Array[x, n, 0].Array[base^# &, n, 0] == number, 
                           And @@ Thread[0 <= Array[x, n, 0] < -base]}, 
                           Array[x, n, 0], Integers]]

usage:
negBase[-10, -1467]
(*
    {{3, 7, 6, 2, 0}}
*)

