Largest Positive Integer

Date: 11/04/2002 at 15:14:07
From: Victoria Szabuniewicz
Subject: Largest positive integers

What is the largest positive integer n for which n^3 + 100 is
divisible by n+10?

I have tried to graph both equations, divide the two equations and 
then graph; then I tried to factor the equations. I have no idea what 
else to do and have come up with nothing but frustration.

Thanks!

Date: 11/06/2002 at 10:47:12
From: Doctor Jubal
Subject: Re: Largest positive integers

Hi Victoria,

Thanks for writing Dr. Math.

You're looking for numbers n such that

  n^3 + 100
 -----------
    n + 10

is an integer.  

If we perform long division on the polynomials, we find

          n^2 - 10n   + 100
        +-----------------------
 n + 10 | n^3 +  0n^2 +  0n + 100
          n^3 + 10n^2
          -----------
              - 10n^2
              - 10n^2 - 100n
              --------------
                        100n + 100
                        100n + 1000
                        -----------
                              - 900

So

  n^3 + 100                       900
  --------- = n^2 - 10n + 100 - -------
    n + 10                       n + 10

And for the original fraction to be an integer, 900 must be divisible 
by n+10. What is the largest value of n such that n+10 divides 900?

Does this help?  Write back if you'd like to talk about this 
some more, or if you have any other questions.

- Doctor Jubal, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 11/06/2002 at 17:07:09
From: Victoria Szabuniewicz
Subject: Thank you (Largest positive integers)

Thank you so much for your help.  It helped me immensely 
and I greatly appreciate it!