Question

What is the largest positive integer \(n\) such that \(n^3+100\) is divisible by \( n+10\)?

Answer 1

n=2 ? Trial and error.

Answer 2

(Just noticed that 12 divides 108)

Answer 3

Oh sorry, the largest:)

Answer 4

Wouldn't there be an endless amount of answers?

Answer 5

No. There is on;y one answer

Answer 6

\(\displaystyle \frac{ n^3+100}{n+10}=\frac{ n^3+1000-900}{n+10}=(n^2-10n+100)-\frac{900}{n+10}\)

Answer 7

so, we want the largest integer \(n\) such that \(900\) divides \(n+10\).

\(\color{red}{n=890}\).

Answer 8

Excellent method

Answer 9

TY:)

Answer 10

Nice problem, I liked it:)