Question

If the number 86 in base ten is represented as 321 in base b, what is the base-ten representation of the number 123 in base b?

Answer 1

When we say \(86\) in base 10 is represented as \(321\) in base \(b\) we mean:$$86=3b^2+2b^1+1b^0$$Notice this is a quadratic equation solvable in \(b\):$$3b^2+2b+1=86\\3b^2+2b-85=0$$Since \(85\) has factors \(1,5,17,85\) it follows \(85\times3\) has factors \(1,3,5,15,17,41,85,255\) and we're interested in two whose difference is \(2\) and whose product is \(85\) -- so focus on \(15,17\):$$3b^2+2b-85=0\\3b^2-15b+17b-85=0\\3b(b-5)+17(b-5)=0\\(b-5)(3b+17)=0$$We clearly see that the only positive solution to the above is \(b=5\) hence we're interested in base \(5\).

Answer 2

Anyways, \(123\) in base \(5\) is then equivalent to the following numeric (decimal) value:$$1b^2+2b^1+3b^0=1(25)+2(5)+3(1)=25+10+3=38$$