# 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

Updated on August 07, 2022

• 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 over 8 years
Disprove just means show a counter example if possible.
@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?
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?