Largest value of $n$

For discussing Olympiad Level Number Theory problems
User avatar
Zzzz
Posts: 172
Joined: Tue Dec 07, 2010 6:28 am
Location: 22° 48' 0" N / 89° 33' 0" E

Largest value of $n$

Unread post by Zzzz » Fri Dec 10, 2010 11:50 am

Find the largest value of $n$ such that $n^3+10$ is divisible by $n+10$.
Every logical solution to a problem has its own beauty.
(Important: Please make sure that you have read about the Rules, Posting Permissions and Forum Language)

User avatar
TIUrmi
Posts: 61
Joined: Tue Dec 07, 2010 12:13 am
Location: Dinajpur, Bangladesh
Contact:

Re: Largest value of $n$

Unread post by TIUrmi » Fri Dec 10, 2010 1:06 pm

\[n^{3} + 10 = (n + 10)(n^{2} - 10n + 100) -990\]
\[n + 10 \| (n + 10)(n^{2} - 10n + 100)\]

So, $n + 10$ should divide $990$
Largest value of $n = 980$
"Go down deep enough into anything and you will find mathematics." ~Dean Schlicter

User avatar
Masum
Posts: 592
Joined: Tue Dec 07, 2010 1:12 pm
Location: Dhaka,Bangladesh

Re: Largest value of $n$

Unread post by Masum » Fri Dec 10, 2010 11:01 pm

I didn't understand the sign $||$ here, because it is used to mean:If $p^a||n$,then $p^{a+1}$ does not divide $n$
However,it was a problem from AIME may be and simialr to my post"find the largest $x$'
$n+10|n^3+10$ and $n^3+100$,then $n+10|990,n_{max}=980$
One one thing is neutral in the universe, that is $0$.

User avatar
TIUrmi
Posts: 61
Joined: Tue Dec 07, 2010 12:13 am
Location: Dinajpur, Bangladesh
Contact:

Re: Largest value of $n$

Unread post by TIUrmi » Sat Dec 11, 2010 12:01 am

That was a typo. Sorry!
"Go down deep enough into anything and you will find mathematics." ~Dean Schlicter

Hasib
Posts: 238
Joined: Fri Dec 10, 2010 11:29 am
Location: খুলনা, বাংলাদেশ
Contact:

Re: Largest value of $n$

Unread post by Hasib » Tue Dec 14, 2010 5:06 pm

Solu ta ki?
A man is not finished when he's defeated, he's finished when he quits.

Hasib
Posts: 238
Joined: Fri Dec 10, 2010 11:29 am
Location: খুলনা, বাংলাদেশ
Contact:

Re: Largest value of $n$

Unread post by Hasib » Tue Dec 14, 2010 6:10 pm

Zzzz wrote:Find the largest value of $n$ such that $n^3+10$ is divisible by $n+10$.
a nice problem!
A man is not finished when he's defeated, he's finished when he quits.

Post Reply