Does c always divide b in number theory divisibility?

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SUMMARY

In number theory, if \( a \) divides \( b \) (denoted as \( a|b \)), then it follows that \( ac = b \) implies \( c \) also divides \( b \). This conclusion is established as a fundamental property of divisibility. The discussion highlights the importance of understanding direct proofs in discrete mathematics, which is essential for deeper mathematical reasoning and problem-solving. The participant also notes personal growth in mathematical skills through auditing a course.

PREREQUISITES
  • Understanding of basic number theory concepts, specifically divisibility.
  • Familiarity with mathematical notation and terminology.
  • Knowledge of direct proofs in discrete mathematics.
  • Experience with mathematical reasoning and problem-solving techniques.
NEXT STEPS
  • Study the properties of divisibility in number theory.
  • Learn about direct proof techniques in discrete mathematics.
  • Explore advanced topics in number theory, such as modular arithmetic.
  • Practice solving problems involving divisibility and proofs.
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Mathematics students, educators, and anyone interested in enhancing their understanding of number theory and mathematical proofs.

chimath35
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If a|b then ac=b; now does c always divide b as well?
 
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chimath35 said:
If a|b then ac=b; now does c always divide b as well?

Of course it does.
 
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Thank you Dick; I did pass the exam I took by the way lol. I did not really get to study much as I was ill, but still did good enough compared to other class mates. I think I am growing A LOT mathematically. I am just now learning how to do direct proofs in discrete math so hopefully I will do better next exam, even though I am just auditing this course. I am learning how to become more of a mathematician I guess.
 

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