Theorem 2.8

Statement: An integer $N$ is divisible by 6 if and only if 2 divides $N$ and 3 divides $N$. That is, $N$ is divisible by 6 if and only if $2 \mid N$ and $3 \mid N$.

This is an "if and only if" (iff) statement, meaning we need to prove both directions:

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

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

Let's break this down step by step.

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

Proof:

Assume $N$ is divisible by 6. By definition of divisibility, this means: $N = 6k \quad \text{for some integer} \, k.$

Since 6 can be factored as $6 = 2 \times 3$, we can rewrite $N$ as: $N = 2 \times 3 \times k.$ Now, observe that:

• $N = 2 \times (3k)$, so $N$ is divisible by 2.
• $N = 3 \times (2k)$, so $N$ is divisible by 3.

Thus, if $N$ is divisible by 6, then $N$ is divisible by both 2 and 3.

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

Proof:

Assume $N$ is divisible by 2 and 3. This means:

• $N = 2m$ for some integer $m$, so $N$ is divisible by 2.
• $N = 3n$ for some integer $n$, so $N$ is divisible by 3.

We need to show that $N$ is divisible by 6.

Since $N$ is divisible by both 2 and 3, it must be divisible by their least common multiple (LCM). The LCM of 2 and 3 is 6. Therefore, $N$ is divisible by 6, i.e.: $N = 6p \quad \text{for some integer} \, p.$

Thus, if $N$ is divisible by both 2 and 3, then $N$ is divisible by 6.

Conclusion:

We have proved both directions:

• If $N$ is divisible by 6, then $N$ is divisible by both 2 and 3.
• If $N$ is divisible by both 2 and 3, then $N$ is divisible by 6.

Thus, the statement is true: $N$ is divisible by 6 if and only if $2 \mid N$ and $3 \mid N$.

---

Would you like further clarification on any of the steps? Or perhaps you'd like to dive deeper into the concepts of divisibility or LCM?

Here are some related questions you may find interesting:

1. How does the concept of the least common multiple (LCM) play a role in divisibility?
2. What is the relationship between divisibility and prime factorization?
3. Can a number be divisible by 2 or 3 but not 6? Give an example.
4. How do we prove divisibility for larger numbers, say 12 or 15?
5. What are some other "if and only if" divisibility theorems, and how are they proved?

Tip: To quickly check if a number is divisible by 6, first check divisibility by both 2 (i.e., even number) and 3 (i.e., sum of digits divisible by 3). If both conditions are satisfied, the number is divisible by 6.