If c|ab and gcd(b, c) = 1 why does c|ac?

  • Context: MHB 
  • Thread starter Thread starter MI5
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Discussion Overview

The discussion revolves around the mathematical statement concerning divisibility, specifically exploring why, given that \( c \) divides \( ab \) and the greatest common divisor of \( b \) and \( c \) is 1, it follows that \( c \) divides \( ac \). The scope includes theoretical reasoning and clarification of definitions related to divisibility.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant presents a theorem stating that if \( c|ab \) and \( \text{gcd}(b, c) = 1 \), then \( c|a \), but expresses uncertainty about why \( c|ac \) follows from this.
  • Another participant attempts to clarify that \( ac = a \cdot c \) by the definition of the divisibility relation.
  • A subsequent participant echoes the confusion about why \( c|ac \) is not immediately clear and requests further explanation.
  • Another participant asks for the definition of the divisibility relation to ensure understanding.
  • A later reply expresses gratitude, indicating some level of satisfaction with the discussion, though it does not clarify the mathematical points raised.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the implications of the theorem and the definition of divisibility. There is no consensus reached on the clarity of the reasoning behind \( c|ac \).

Contextual Notes

The discussion highlights a potential gap in understanding the definitions and implications of the divisibility relation, as well as the assumptions underlying the theorem presented.

MI5
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Theorem: If c|ab and (b, c) = 1 then c|a.

Proof: Consider (ab, ac) = a(b, c) = a. We have c|ab and clearly c|ac so c|a. It's not so clear to me why c|ac. Perhaps I'm missing something really obvious.
 
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MI5 said:
It's not so clear to me why c|ac.
This is because ac = a * c, i.e., by the definition of the | (divides) relation.
 
Evgeny.Makarov said:
This is because ac = a * c, i.e., by the definition of the | (divides) relation.
I still don't understand I'm afraid. Could you say bit more please?
 
MI5 said:
I still don't understand I'm afraid. Could you say bit more please?
I need to be sure you know the definition of the relation denoted by |. Could you write this definition?
 
WOW! All I can say is thanks. (Giggle)
 

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