Final answer:
A number n divisible by both 2 and 3 means there are integers k and m such that n=2k and n=3m. Since both expressions define n, setting them equal shows that k must be divisible by 3, leading to n=6l, proving n is divisible by 6.
Step-by-step explanation:
To prove that a number n is divisible by 6 if it is divisible by both 2 and 3, we first need to understand the concept of divisibility. A number is divisible by another if it can be divided by that number without leaving a remainder. If n is divisible by 2, there exists an integer k such that n = 2k. Similarly, if n is divisible by 3, there exists an integer m such that n = 3m. Now, since n is divisible by both, it means that 2k = 3m, which implies that k is divisible by 3. Therefore, k = 3l for some integer l, which leads us to n = 2(3l) = 6l. Thus, n is divisible by 6, as there is an integer l such that n = 6l. This proves that if a number is divisible by both 2 and 3, then it is necessarily divisible by 6, the least common multiple of 2 and 3.