How do i prove the question below?
"Prove that a^2|b and b^3|c then a^4b^5|c^3."
I would even say that under this assumption, a^8b^5 | c^3. You prove this in a way that is similar to how you prove the following fact. Suppose 2 boxes of size A fit into a box of size B and 3 boxes of size B fit into a box of size C. Then 4 boxes of size A and 5 empty boxes of size B fit into 3 boxes of size C.
Let a^2 | b and b^3 | c. Then there exists integers m and n such that b = ma^2 and c = nb^3. We need to show that a^4 b^5 | c^3 i.e. there exists an integer k such that c^3 = ka^4 b^5. Note that
c^3 = (nb^3)^3
c^3 = n^3 b^9
c^3 = n^3 b^5 * b^4
c^3 = n^3 b^5 * (ma^2)^4
c^3 = n^3 b^5 * m^4 a^8
c^3 = (n^3 m^4 a^4)(a^4 b^5)
Set k = n^3 m^4 a^4. Then c^3 = k * a^4 b^5. Then a^4 b^5 | c^3 as needed to show.