SOLUTION: When the number 𝑛 is written in base 𝑏, its representation is the two-digit number 𝐴𝐵 where 𝐴=𝑏−2 and 𝐵=2. What is the representation for 𝑛 in base (𝑏

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: When the number 𝑛 is written in base 𝑏, its representation is the two-digit number 𝐴𝐵 where 𝐴=𝑏−2 and 𝐵=2. What is the representation for 𝑛 in base (𝑏      Log On


   

Question 1158010: When the number 𝑛 is written in base 𝑏, its representation is the two-digit number 𝐴𝐵 where 𝐴=𝑏−2 and 𝐵=2. What is the representation for 𝑛 in base (𝑏−1)?
Answer by KMST(5377) About Me  (Show Source):
You can put this solution on YOUR website!
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%5E3%2BBx%5E2%2BCx%2BD .
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%2A10%5E2%2B2%2A10%2B4 .

The value of the number n written as AB in base b is
n=Ab%2BB, and we know that b%3EA=b-2 and b%3EB=2 .
So, b%3E=3 and b-1%3E=2 .
Substituting b-2 for A and 2 for B, we get
n=%28b-2%29b%2B2
Now we are dealing with polynomials.
n=%28b-2%29b%2B2-->n=b%5E2-2b%2B2-->n=b%5E2-2b%2B1%2B1-->n=%28b-1%29%5E2%2B1-->Highlight%28n=1%2A%28b-1%29%5E2%2B0%2A%28b-1%29%2B1%29
and that polynomial in %28b-1%29 is the value of the number that in base b-1 would be written as highlight%28101%29 .