Expert: Paul Klarreich - 8/26/2011

Question
Here's my question about function:

  Let F(x#=x^99+K,

Part A:When F#x)is divided by x+1,the remainder is 1. Find the value of K.
Part B:Hence,find the remainder when 9^99 is divided by 10.

PS:Well,i had actually solved the Part A of the question as the value of K is (-1)^99+K=1 and the answer is 2.But i was then freaking stuck on Part B.Are there any connection in between?

Answer
Questioner:Desmond
Country:Hong Kong
Category:Advanced Math
Private:No
Subject:Function

Question:
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I have attempted to fix the notation:

Here's my question about function:

 Let F(x) = x^99 + K,

Part A:When F(x)is divided by x+1,the remainder is 1. Find the value of K.

>>>>>>>>>>> I assume you used the Remainder Theorem, and wrote:

F(-1) = 1 = (-1)^99 + K

Part B:Hence,find the remainder when 9^99 is divided by 10.

Write:  G(x) = (x - 1)^99

G(10) = (10 - 1)^99.

When you expand (x - 1)^99, you obtain a polynomial with 100 terms.  All but the last term have at least one factor of x.

So if you expand (10 - 1)^99, you obtain a polynomial with 100 terms.  All but the last term have at least one factor of 10.  

That last term is (-1)^99 = -1.  That is the remainder. (well, actually, you are not supposed to say that -- the remainder is actually 9.)

As a check, note that:
A.  9^2 = 81.
B.  9^2n = 81^n = some number ending in 1.
C.  9^(2n+1) = 81^n * 9 = some number ending in 9.

PS:Well,i had actually solved the Part A of the question as the value of K is
(-1)^99+K=1 and the answer is 2.But i was then freaking stuck on Part B.Are
there any connection in between?

>>>>>>>> I am not sure.  Probably your teacher has some cute idea in mind.