# Fix any errors in my proof (beginner)

**Theorem.** *For every integer n, $6 \vert n$ $\ \text iff$ $2 \vert n$ and $3 \vert n$. Proof. *Let $n$ be an arbitrary integer.

$(\rightarrow)$ Suppose $6 \vert n$. $\exists k \in \mathbb{Z}(6k = n)$, so $3(2k) = n$ and $n \vert3$. Likewise, $6k = 2(3k) = n$, so $2 \vert n$. Thus, $2 \vert n$ and $3 \vert n$.

$(\leftarrow)$ Suppose $n \vert 3$ and $n \vert 2$. $\exists j \in \mathbb{Z} (3j = n)$ and $\exists k \in \mathbb{Z} (2k = n)$. Thus $6(k - j) = 6k - 6j = 3(2k) - 2(3j) = 3n - 2n = n$. Becaus $k - j$ is an integer, it follows that $n \vert 6$.

$\therefore$ $\forall n \in \mathbb{Z}$, $6 \vert n$ $\ \text iff$ $2 \vert n$ and $3 \vert n$.

## Answer

**Answers can be viewed only if**

- The questioner was satisfied and accepted the answer, or
- The answer was disputed, but the judge evaluated it as 100% correct.

The answer is accepted.

- answered
- 76 views
- $5.00

### Related Questions

- proof by induction
- Proof through inclusion (A∆B) ∪ A = A ∪ B
- Two statistics proofs with regressions, any help much appreciated!
- Fix any errors in my proof (beginnner)
- Prove that: |x| + |y| ≤ |x + y| + |x − y|.
- Topic: Large deviations, in particular: Sanov's theorem
- Let $f\in C (\mathbb{R})$ and $f_n=\frac{1}{n}\sum\limits_{k=0}^{n-1} f(x+\frac{k}{n})$. Prove that $f_n$ converges uniformly on every finite interval.
- Fix any errors in my proof (beginner)