find x if 283(x)=A2(16)-153(80)+100110(2) discrete mathematics

Question

i really ned help with this problem i'm in discrete mathematics and i don't know where to begin

Answer 1

Hi James, this is a challenging problem. Let's put the right-hand side into base-10.

A2 in base 16 = 10*16 + 2 = 162 in base 10.

153 in base 8 = 1 * 8^2 + 5*8 + 3 = 107 in base 10.

100110 in base 2 = 2^5 + 2^2 + 2 = 38 in base 10.

So the expression is 162 - 107 + 38 = 93.

Now we can write an equation: if I want to convert 283 from base x to base 10, I write:

2x^2 + 8x + 3 = 93.

So 2x^2 + 8x + 90 = 0.

Get rid of the 2 to get x^2 + 4x + 45 = 0.

Factor to get (x + 9)(x - 5) = 0.

So the possible values for x are -9 and 5; since it must be positive to be a valid base, the answer is 5.