How does these two statements imply each other?

Question

How does these two statements imply each other?

For integer $n$: '16 divides $n^3$ implies $4$ divides $n$.'

For integer $n$: '$4$ divides $n$ implies 16 divides $n^3$.'

I think the logical statement is:

For integer $n$: '$4$ divides $n$ is a necessary and sufficient condition 16 divides $n^3$.'

Thanks in advance.

Answer 1

If $4 \mid n$ then $n = 4k$ for some $k$. Thus $n^3 = 64k^3 = 16(4k^3)$, so that $16 \mid n^3$.

On the other hand, if $16 \mid n^3$, then $2 \mid n^3$. Since 2 is a prime, we must have $2 \mid n$. Thus $n$ is of the form $4k$ or $4k+2$. If $n = 4k+2$ for some $k$, then $n^3 = 64k^3 + 96k^2 + 48k + 8$ which is not divisible by 16, a contradiction. Hence $4 \mid n$

Answer 2

If the highest power of any prime $p$ in $n$ is $r,$

then the highest power of $p$ in $n$ will be $3r$

So, if $\displaystyle16=2^4$ divides $n^3,3r\ge4\iff r\ge2$

If $\displaystyle4=2^2$ divides $n, r\ge 2\implies 3r\ge6$

Answer 3

Let $\,n = 2^{\large k}m,\ m$ odd. Then $\ 2^{\large \color{#c00}4} \mid n^{3}\!= 2^{\large\color{#c00}{3k}} m^3 \!\!\!\!\color{blue}{\overset{\ \rm UF}\iff}\! \color{#c00}{4\le 3k} \!\iff\! \color{#0a0}{2\le k} \!\!\color{blue}{\overset{ \rm UF}\iff}\! 2^{\large\color{#0a0}2}\mid 2^{\large\color{#0a0}k}m = n.$

The arrows $\!\color{blue}{\overset{ \rm UF}\iff}\!$ use uniqueness of prime factorization. Or, more simply, we may employ parity as follows: $ $ if $\,j\,$ is odd then $\,2\mid ij\iff 2\mid i.\,$ So $\,2^k\mid ij \iff 2^k\mid i\,$ follows by induction (this implies the uniqueness of factorizations of the form $\,2^k m\,$ for $\,m\,$ odd).