Is 1/5*n^5 + 1/3*n^3 + 7/15*n Always an Integer for All n?

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SUMMARY

The expression 1/5*n^5 + 1/3*n^3 + 7/15*n is proven to be an integer for all integers n. The proof involves analyzing the expression 3n^5 + 5n^3 + 7n under modulo 5 and modulo 3. This approach confirms that the expression yields integer results regardless of the integer value of n. The discussion emphasizes the importance of modular arithmetic in validating integer properties of polynomial expressions.

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  • Understanding of polynomial expressions and their properties
  • Knowledge of modular arithmetic, specifically modulo 5 and modulo 3
  • Familiarity with integer proofs and mathematical induction
  • Basic algebraic manipulation skills
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  • Explore polynomial identities and their proofs
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dessy
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Prove that 1/5*n^5+1/3*n^3+7/15*n is an integer for all integers n.
 
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Look at 3n^5 + 5n^3 + 7n mod 5 and mod 3.
 
Thanks, Morphism
 

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