A Bit of Math

CLAIM: If "a divides b" (a|b) and "b divides c" (b|c), then "a divides c" (a|c)

Example:
2 divides 4 (4/2) and 4 divides 8 (8/4) . So, 2 divides 8 (8/2).

You might go "duh" but can you prove it to me that it is true ALWAYS? XD

PROOF
If a|b, there exists an integer q such that b=qa. [Eq 1](from the Division Algorithm)

If b|c, there exists an integer r such that c=br [Eq 2](from the Division Algorithm)
Substituting b from Eq 1 into Eq 2, we obtain

c=rqa

We notice that 'rq' are integers and can be represented as Q (a result of multiplying two integers together is an integer)

Therefore, c=Qa

From the Division Algorithm, a|c. [Proven]

Note: Division Algorithm states that if 'a' and 'b' are integers and 'a' divides 'b', there exists an integer 'q' such that b=qa.