Question 1158010
The value of a number in any base is really a polynomial.
If ABCD is how a number is written in base {{{x}}} , its value is
{{{Ax^3+Bx^2+Cx+D}}} .
Of course, if ABCD is how a number is written in base {{{x}}} ,
we know that {{{x}}} is an integer,
that {{{x}}} is greater than A, B, C, and D,
and that {{{x}}} would be written as 10 in base {{{x}}}
The value of number 324 in base 10 is {{{3*10^2+2*10+4}}} .
 
The value of the number {{{n}}} written as {{{AB}}} in base {{{b}}} is 
{{{n=Ab+B}}}, and we know that {{{b>A=b-2}}} and {{{b>B=2}}} .
So, {{{b>=3}}} and {{{b-1>=2}}} .
Substituting {{{b-2}}} for {{{A}}} and {{{2}}} for {{{B}}}, we get
{{{n=(b-2)b+2}}}
Now we are dealing with polynomials.
{{{n=(b-2)b+2}}}-->{{{n=b^2-2b+2}}}-->{{{n=b^2-2b+1+1}}}-->{{{n=(b-1)^2+1}}}-->{{{Highlight(n=1*(b-1)^2+0*(b-1)+1)}}}
and that polynomial in {{{(b-1)}}} is the value of the number that in base {{{b-1}}} would be written as {{{highlight(101)}}} .