Question

Prove the following statement using a direct proof: If a,b,, and c are integers such that b is a multiple of a³ and c is a multiple of b³, then c is a multiple of a³.

Answer 1

To prove the statement using a direct proof, assume that b is a multiple of a³ and c is a multiple of b³. Substitute the values of b and c in terms of a into the equation. Simplify the equation to show that c is also a multiple of a³.

Step-by-step explanation:

To prove the statement using a direct proof, we need to show that if b is a multiple of a³ and c is a multiple of b³, then c is a multiple of a³. Let's assume that b is a multiple of a³ and c is a multiple of b³. Since b is a multiple of a³, we can write b = ka³, where k is an integer. Similarly, since c is a multiple of b³, we can write c = mb³, where m is an integer. Now, we can substitute the value of b in terms of a and k in the equation for c: c = m(ka³)³ = m(k³a⁹) = (mk³)a⁹. Since mk³ is an integer, we can see that c is also a multiple of a³. Therefore, our assumption holds true and we have proved that if b is a multiple of a³ and c is a multiple of b³, then c is a multiple of a³.