If $n$ is divisible by 6, then $n$ is divisible by 3

Question

There are various questions that are asking me to find out which of the following statements are true and to explain briefly (no proof required, although id be interested to see what it was). I've tried finding similar proofs online that could at least point me in the right direction as to figuring out which are true and false but didn't find anything.

1. If $n$ is divisibale by 6, then $n$ is divisible by 3

2. If $n$ is divisible by 3 then $n$ is divisible by 6.

3. If $n$ is divisible by 2 and $n$ is divisible by 3, then $n$ is divisible by 6

How should I approach a problem like this? (This is my first class on mathematical reasoning)

Answer 1

For the first question, you can prove it like this. Suppose that $n$ is divisible by $6$. Then, $n = m*6$ where $m$ is an integer. Note that $6 = 2*3$ so that $n = m*(2*3) = (2m)*3.$ Therefore, $n$ is also a multiple of $3$.

Also, remember that to show something is false, you just need to provide one counterexample. This would apply to question 2.

Answer 2

Partial answer:

3) $n$ is divisible by $3$ and $2$ , then $n$ is divisible by $6$.

$2|n$ implies $n=2k$;

Euclid's lemma: If $p$, prime, divides $ab$, then $p$ divides $a$ or $p$ divides $b$.

$n=2k$; Also: $3$ divides $n=2k$;

Euclid's lemma: $3| k$, i.e. $k=3l$;

Combining :

$n= 2k=2(3l)=(2)(3)l=6l$,

hence $6$ divides $n$.