# Math Help - Proof of divisibility

1. ## Proof of divisibility

Let n be an odd integer. Prove that n^3/n is divisible by 24.

2. $\frac{n^3}{n} = n^2$, which is not necessarily divisible by 24 if n is odd. Examples: $3^2 = 9, 5^2 = 25$. Perhaps you wrote the problem incorrectly?

3. OOps! Yes I did. The correct one is: Prove (n^3)-n is divisible by 24.

4. Ok. $n^3 - n = n(n^2 - 1) = n(n+1)(n-1)$.

Since the product is composed of three consecutive integers, one of them is divisible by 3. Also, n-1 and n+1 are consecutive even integers. Among every two consecutive even integers, one of them is divisible by 4, and the other is divisible by 2. Hence the two multiplied together are divisible by 8. Since 8 and 3 are both factors of the product and they are relatively prime, the product is divisible by 24.

5. Thank you so much Icemanfan!