Question

Let n∈Z. Prove that 3∣ n³-n.

Answer 1

To prove that 3 divides n³ - n for any integer n, we can use factorization and properties of consecutive integers. By factoring the expression n³ - n, we can show that it is divisible by both 2 and 3, implying that it is divisible by 6. Thus, we can conclude that 3 divides n³ - n.

Step-by-step explanation:

To prove that 3 divides n³ - n for any integer n, we can start by factoring out n from the expression:

n³ - n = n(n² - 1)

Now, we can factor the expression further:

n³ - n = n(n - 1)(n + 1)

Since n, n - 1, and n + 1 are consecutive integers, one of them must be divisible by 2. Moreover, among three consecutive integers, at least one must be divisible by 3. Therefore, n(n - 1)(n + 1) is divisible by both 2 and 3, which means it is divisible by 6. So, we conclude that 3 divides n³ - n for any integer n.