Def: An interger "m" divides an integer "n" if there is an integer "q" such that n=mq.
?Suppose a, b, and c are integers such that a divides b and b divides c. Prove that a divides c.
If b is an integer factor of b, and b is an integer factor of x, then
b = ma and c = nb, where m and n are both integers.
It follows that c = m n a. Since m and n are both integers, so is mn. Therefore a is an integer factor of c, or, in other words, "a divides c".
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.