Proving with Congruence of intergers

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SUMMARY

The discussion centers on proving that the expression b³ + b² + 1 is not divisible by 5. The user asserts that b³ + b² + 1 is not congruent to 0 modulo 5 and contemplates using proof by contradiction. The conversation suggests evaluating the expression for all integer values of b from 0 to 4 modulo 5 to confirm the conclusion. This method effectively demonstrates the non-divisibility of the polynomial by 5.

PREREQUISITES
  • Understanding of modular arithmetic, specifically modulo 5.
  • Familiarity with polynomial expressions and their properties.
  • Knowledge of proof techniques, particularly proof by contradiction.
  • Basic skills in evaluating congruences.
NEXT STEPS
  • Explore the concept of modular arithmetic in greater depth.
  • Learn about proof by contradiction and its applications in number theory.
  • Investigate polynomial congruences and their divisibility properties.
  • Practice evaluating expressions modulo n for various integers.
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This discussion is beneficial for students of mathematics, particularly those studying number theory, as well as educators looking for examples of proof techniques and modular arithmetic applications.

doggie_Walkes
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It just how do i prove that

b^3 +b^2 +1 does not divide by 5


Im thinking this way,
cause i know that b^3 +b^2 +1 is not congruent to 0(mod5)

therefore we use contradition to prove it. I am just not sure how to use contradition? or maybe I am looking at this in a completely bad light? maybe there is another method?
 
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This can be worked out by considering all the cases from 0 to 4 mod 5.
 

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