Prove the following using a direct proof. Your proof should be expressed in com-plete English sentences. If a, b, and c are integers such that b is a multiple of a3 and c is a multiple of b2, then c is a multiple of a6.

Proved

Step-by-step explanation:

Given

Integers: a, b, c

Required

Show that c is a multiple of a⁶

First, we need to list out the multiples of a³

Since b is a multiple of a³, then b can be any of the listed multiples

Next, is to list out the multiples of b²

Since c is a multiple of b², then c can be any of the listed multiples

To list out the multiples of a⁶, we have to get the common multiples of a³ and b² in terms of a

Substitute a³ for b in

Recall that c is a multiple of b²

From the above listed multiples, we have a⁶ listed as one of the multiple; Hence, c is a multiple of a⁶

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