the question is consider the statement: if a/b and b/c then a/c. Give a direct proof?

Answer 1

if a = b and b=c, then a has to be the same as c.
i.e.
a = b = c

Answer 2

By a/b, I am going to take it to mean "a divides b".

Proof:

Assuming a divides b. It then follows that

a = bj, for some integer j.

Assume that b divides c. Then,

b = ck, for some integer k.

Substituting in b = ck for a = bj, we get

a = (ck)j, OR
a = c(jk).

Since jk is an integer, it follows that

a divides c.

Answer 3

If a|b, then, it exists an integer m/ b=ma

If b|c, then, it exists an integer p/ c = pb

Since c = pb = p (ma) = (pm) a, then it exists an integer j= pm, so that c = ja. Hence, a|c

Ana

Answer 4

Because if a = b and b=c, then a has to be the same as c.
a=b=c