Question

in what base b is 1030(base b) divided by 4(base b) = 206(base b)?

Answer 1

@dan815

Answer 2

$$1030_b=0\cdot b^0+3\cdot b^1+0\cdot b^2+1\cdot b^3=3b+b^3\\4_b=4\\206_b=6+2b^2$$

Answer 3

so $$\frac{3b+b^3}4=6+2b^2\\3b+b^3=24+8b^2\\b^3-8b^2+3b-24=0\\b^2(b-8)+3(b-8)=0\\(b^2+3)(b-8)=0\implies b=8$$

Answer 4

the more intuitive and quick way of solving is to notice that dividing by 4 is the same as shifting the digits over by 1 and multiplying by 2. shifting the digits over by one in base \(b\) corresponds to dividing by \(b\) in the same way that dividing, say, 260 by 10 yields 26. this means that \(\dfrac14=\dfrac1b\cdot2\implies b=8\)

Answer 5

thank you so much! :)