Prove or disprove: If $a\mid (b+c)$, then either $a\mid b$ or $a\mid c$

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Solution 1

HINT. Maybe try a few examples, like $a=1,b=2,c=3$, and $a=2,b=4,c=6$, and $a=5,b=2,c=3$.

So does this work all the time? If it does how would you show it? If it does not, give an example where it fails. And if it fails, don't stop there! When does it fail? Is it sometimes true? For what examples? Can you add more assumptions to make the statement always true?

"Don't just read it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? Where does the proof use the hypothesis?" - Paul Halmos

Solution 2

Try $b + c = a, 1\leq b,c<a$

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Lil
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Lil

Updated on August 07, 2022

Comments

  • Lil
    Lil less than a minute

    Prove or disprove: If $a\mid (b+c)$, then either $a\mid b$ or $a\mid c$

    I'm so confused how to go about this since it says prove or disprove. Should I start off by doing proof by contradiction? Step by step explanation please?

    • John Habert
      John Habert over 8 years
      Disprove just means show a counter example if possible.
    • Admin
      Admin over 8 years
      First, you should decide if you think it's true or not.
    • JB King
      JB King over 8 years
      How about a=6, b=3, c=9?
  • Lil
    Lil over 8 years
    ok so I plugged in the values and it fails for the last example so I have to disprove the statement. What theorems should I use? Sorry I'm really confused with number theory..
  • mathematics2x2life
    mathematics2x2life over 8 years
    @Lil But by trying an example you have disproven the statement! The statement says this should be true for all natural numbers (I assume natural) but you just found a few where it did not work. So the statement is false and you are done. But as I suggested, it's good for learning not to stop there. See if you can find examples on your own of where it is false, get to the heart of why it is false. Find more true examples. Can you add assumptions on $a$ for example to make the statement true?
  • Lil
    Lil over 8 years
    my teacher told us that examples don't count as a full answerfor proofs. We have to incorporate theorems and generalize in order order for it to be a formal proof. Is there a formal proof for this question?
  • mathematics2x2life
    mathematics2x2life over 8 years
    There can be no proof for statements which are incorrect, because they are incorrect! All that is needed to disprove them is a single case. What your professor probably means is that examples do not count for proofs of correct statements (as the Jewish proverb goes, 'For example is not a proof'). So there is no proof. However, you can give several examples where it fails and explain why it fails (for example, if $a \mid (b+c)$ then $a \leq b+c$. But does that mean $a <b$ and $a<c$? So does that mean for the statement? Can you back that statement up with an example, maybe one you gave?