# If $ab | c$, does $a | c$?

Basically, prove or disprove that $ab|c \implies a|c$.

It seems like it should, but I don't know how to go about proving it.

• If $ab \mid c$ then we can write $c = abn$ for some integer $n$. Now let us add some suggestive parentheses: $c = a(bn)$. What can you conclude? – Bungo Mar 25 '16 at 1:53
• just write down the definition of dividing. – Melquíades Ochoa Mar 25 '16 at 1:55

ab|c means there exists an integer d such that (ab)d=c

So consider the integer bd now

a(bd)=c hence a divides c

Hint: Doesn't "divides" really mean "is a factor of"? If so, then aren't factors of the divisor also factors of the dividend?

Example: $100$ is a factor of $1000$. Because $100=4\times 25$, you immediately know that $4$ is a factor of $1000$, and $25$ is a factor of $1000$.

Hint:

Divisibility is a transitive relation.

By definition, $ab|c \Leftrightarrow c = abk$ for some $k \in \mathbb{Z}$. Similarly, $a|c \Leftrightarrow c = am$ for some $m \in \mathbb{Z}$. Let $m = bk$ and you are done.