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If $a\mid c$ and $b\mid c$, must $ab$ divide $c$? Justify your answer.

$a\mid c$, $c=ak$ for some integer $k$

$b\mid c$, $c=bu$ for some integer $u$

From here I wanted to try to check if there were counter examples I could use,

$c\ne(ab)w$ for some integer $w$

From here I got stuck because there is nothing I can plug into that equation so I know that I am probably missing something.

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    $2,4,4 \qquad{}$ – 2017-01-29
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    $4\mid 12$ and $6\mid12$ but $4\times6\nmid12. \qquad$ – 2017-01-29
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    Fun fact: If $a\mid c$ and $b\mid c$ for coprime $a$ and $b$, then $ab\mid c$. – 2017-01-29

3 Answers 3

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This is not true.

Take for instance $a=b=2$ and $c=2$.

Then $a\mid c $ and $b\mid c$, but

$$ab=4\nmid 2=c.$$

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$4\mid 12$ and $6\mid12$ but $4\times6\nmid12. \qquad$

The proposition is true when $\gcd(a,b)=1.$

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    Womp. This was the example I was going to use. $+1$! – 2017-01-29
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No it must not,you can give a simple counter-example like $3\mid 9$ and $9\mid 9$ but obviously $$ab=9\cdot3=27 \nmid9$$ Hope it helps you out!

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    $a = 3, b = 5, c=5, gcd(a,b) = 1$ but the $ab \not \mid c$ – 2017-01-29
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    True that, ill change this up in a second, I was thinking of an extension to Euclid's Lemma – 2017-01-29