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I got stuck on this question right after previous ones went quite well.

I know that by definition of divisilibity, due to $a\mid bc:$

$$bc = ak$$

(where $k$ is some integer)

But I already get stuck here. In previous ones, I could assume that the second part of the statement was true and then through substituting and algebra could work it out, but the 'or' in it is throwing me off.

I tried assuming for example that $a\mid b$ is true. After

$$b = am$$

(where $m$ is some integer)

Substituting:

$$amc = ak$$

I just end up going nowhere.

How do I deal with this?

Bill Dubuque
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Tiisje
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1 Answers1

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What you are trying to prove is false. To see this, consider $6\mid 2\times 3$. Here $6\nmid 2$ and $6\nmid 3$.

Shaun
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  • I did find that out by looking for a counterexample later on, but if that is the case, shouldn't the attempt to prove a false statement give a clue that the statement is false? For example when coming across a contradiction. Counterexamples might be easy to find for a question like this, but what if a counterexample is not intuitively obvious? – Tiisje Nov 22 '21 at 16:30
  • It depends on the question, @Tiisje. There is no one-size-fits-all approach to number theory questions. – Shaun Nov 22 '21 at 16:33
  • I understand that, it would be nice if one algorithm could solve every problem, but does that mean that the approach I took (trying to find hints that a statement is false by attempting to prove it) happens to be non-viable for this specific question? Or is that in general just a bad method to rely on? – Tiisje Nov 22 '21 at 16:38
  • Well, if you're looking for a general tip, @Tiisje, I would say that trying out simple examples of the problem at hand is essential. Not only will that weed out easy counterexamples, but it would give you some intuition about whether the statement is true or false, and why. – Shaun Nov 22 '21 at 16:42
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    Alright, well thanks for your time and the tip, I'll keep it in mind. – Tiisje Nov 22 '21 at 16:44
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    Please strive not to add more (dupe) answers to dupes of FAQs, cf. recent site policy announcement here. – Bill Dubuque Nov 22 '21 at 17:59