Difference between revisions of "1986 AIME Problems/Problem 5"
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== Problem == | == Problem == | ||
| − | What is that largest positive integer <math>n</math> for which <math>n^3+100</math> is divisible by <math>n+10</math>? | + | What is that largest [[positive integer]] <math>n</math> for which <math>n^3+100</math> is [[divisible]] by <math>n+10</math>? |
== Solution == | == Solution == | ||
| − | If <math>n+10 \mid n^3+100</math>, <math>gcd(n^3+100,n+10)=n+10</math>. Using the [[Euclidean | + | If <math>n+10 \mid n^3+100</math>, <math>\gcd(n^3+100,n+10)=n+10</math>. Using the [[Euclidean algorithm]], we have <math>\gcd(n^3+100,n+10)= \gcd(-10n^2+100,n+10)= \gcd(100n+100,n+10)= \gcd(-900,n+10)</math>, so <math>n+10</math> must divide 900. The greatest [[integer]] <math>n</math> for which <math>n+10</math> divides 900 is 890; we can double-check manually and we find that indeed <math>900 \mid 890^3+100</math>. |
== See also == | == See also == | ||
* [[1986 AIME Problems/Problem 4 | Previous Problem]] | * [[1986 AIME Problems/Problem 4 | Previous Problem]] | ||
* [[1986 AIME Problems/Problem 6 | Next Problem]] | * [[1986 AIME Problems/Problem 6 | Next Problem]] | ||
* [[1986 AIME Problems]] | * [[1986 AIME Problems]] | ||
| + | |||
| + | [[Category:Intermediate Number Theory Problems]] | ||
Revision as of 11:30, 29 January 2007
Problem
What is that largest positive integer 
for which 
is divisible by 
?
Solution
If 
, 
. Using the Euclidean algorithm, we have 
, so must divide 900. The greatest integer for which divides 900 is 890; we can double-check manually and we find that indeed 
.
