Find all integers n for which this fraction is an integer

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SUMMARY

The discussion focuses on identifying all integers n for which the expression (n^3 + 2010) / (n^2 + 2010) results in an integer. Participants quickly identify 0 and 1 as straightforward solutions. Further exploration reveals that no integer satisfies the equation n^3 + 2010 = 0. For those with programming skills, implementing a brute-force algorithm is suggested as a viable method to uncover additional solutions.

PREREQUISITES
  • Understanding of polynomial expressions and integer properties
  • Basic knowledge of algebraic manipulation
  • Familiarity with programming concepts for algorithm implementation
  • Experience with brute-force search techniques
NEXT STEPS
  • Research polynomial long division to analyze rational expressions
  • Learn about integer factorization methods for deeper insights
  • Explore programming languages like Python for implementing brute-force algorithms
  • Study number theory concepts related to divisibility and integer solutions
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Mathematicians, computer science students, and anyone interested in solving integer equations or enhancing their algorithmic problem-solving skills.

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Find all integers n for which the fraction n ^ 3 + 2010 / (n ^ 2 + 2010) is equal to integer.

please I need help :( Thank you
 
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Which values immediately come to your mind?
 


0 and 1
 


Right, those are the only two that are easy to find. Another possibility would have been to find an integer such that n^3 + 2010=0 but there are none. If you have any programming skill you could try and brute force it with a relatively easy to implement algorithm.
 

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