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### 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?

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!
```
Associated Topics:
High School Polynomials

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