To prove that a divides c, we need to show that there exists an integer q such that c = aq.
Given that a divides b, we know that there exists an integer p such that b = ap. Similarly, since b divides c, there exists an integer r such that c = br.
Now, we can substitute the value of b in terms of a from the first equation into the second equation:
c = (ap)r
Using the associative property of multiplication, we can rewrite this as:
c = a(pr)
Since p and r are both integers, pr is also an integer. Therefore, we can let q = pr, and we have:
c = aq
Which proves that a divides c.
In this proof, we used the given information that a divides b and b divides c to construct a chain of equalities and substitutions that ultimately demonstrated that a divides c.