[Math] Transitivity of the divisibility relationship among integers.

divisibilityproof-verification

Prove if a, b and c are integers such that a is divisible by b, and b is divisible by c, then a is divisible by c
I've recently been looking into divisibility proofs since I'm new to proofs as a whole, and have come across one particular question that is throwing me off. I'm not sure whether it's throwing me off because it's a false proof (in which case I cannot find a counter-example) or whether my method is wrong.

"Prove if true or disprove through counter-example: if a, b and c are integers such that a is
divisible by b, and b is divisible by c, then a is divisible by c."

My methodology involves the following…

a/b = k
b/c = z
a/c = n

Where k, z and n are all integers.

Thus…
b = a/k
c = b/z
a = czk

So…

a/c = czk/(b/z)

So…

a/c = k/z

But I've hit a wall there. Seems very messy and I doubt I'm on the right track. Would appreciate any help! Thanks!

Best Answer

Use this definition of divisibility:

$a$ divides $b$ if and only if $b=k\cdot a$ for some $k\in\mathbb N$.

Here is an outline of what you should do:

  1. Write down what it means that $a$ divides $b$. (giving you equation (i))
  2. Write down what it means that $b$ divides $c$. (giving equation (ii))
  3. Write down what it means that $a$ divides $c$ (which is what you want to prove)
  4. Try to merge equations (i) and (ii) to get something that can be used for (3).
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