Prove or disprove the statement “An integer n is divisible by 12 if and only if $n^3$ is divisible by 12.”

If integer n is divisible by 12, for some integer m:

$$\begin{aligned} n & = 12m \\ n^3 & = 12^3m^3 \end{aligned}$$

Clearly, $n^3$ is divisible by 12. How about the converse ? Is n divisible by 12 if $n^3$ is divisible by 12 ? Note $12 = 2^2 \times 3$, so one counter example would be:

$$\begin{aligned} n^3 & = 2^3 \cdot 3^3 = 12 \cdot 18 \\ n & = 2 \cdot 3 = 6 \end{aligned}$$

In other words, $n^3$ is divisible by 12, but not n. Hence the converse is false.

# posted by rot13(Unafba Pune) @ 9:16 PM