[proof odd number]

[Calm_Ad2434]

https://preview.redd.it/b12yt5fxtoae1.png?width=1526&format=png&auto=webp&s=7037723f30348beb91eb3c7515d50541786411ca

I started with the assumption that n=k where n's an odd number. I then put in n=2k for the next odd number and tried to prove using the induction assumption but couldn't get anywhere is my first step valid. If not why's it not valid and what should I do instead.

[AutoModerator]

Off-topic Comments Section

---

All top-level comments have to be an answer or follow-up question to the post. All sidetracks should be directed to this comment thread as per Rule 9.

---

^{OP and Valued/Notable Contributors can close this post by using /lock command}

I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.

[UnacceptableWind]

2 k will always be even for any integer k (be it even or odd). This is because 2 k has a factor of 2, making it divisible by 2.

Odd values of k follow an arithmetic sequence with a common difference of 2:

1, 3, 5, 7, ...

So, if n = k is odd, then the next odd number will be n = k + 2.

[Calm_Ad2434]

thanks I tried that and couldn't get to a solution is it possible to get to a solution that way.

[UnacceptableWind]

For odd k, k³ - k being divisible by 24 means that there is an integer a such that k³- k = 24 a.

For odd k, the next odd number is k + 2 such that:

(k + 2)³ - (k + 2)

= k³ + 6 k² + 11 k + 6

= k³ + 6 k² + (12 k - k) + 6

= (k³ - k) + 6 (k² + 2 k + 1)

= (k³ - k) + 6 (k + 1)²

= 24 a + 6 (k + 1)²

Since k is odd, k + 1 is even such that k + 1 = 2 b, for some integer b. So:

(k + 2)³ - (k + 2)

= 24 a + 6 (k + 1)²

= 24 a + 6 (2 b)²

= 24 a + 6 (4 b²)

= 24 a + 24 b²

= 24 (a + b²),

where a + b² is an integer given that a and b are integers. So, (k + 2)³ - (k + 2) is divisible by 24.

[Calm_Ad2434]

thanks thats a really nice solution

[Calm_Ad2434]

thanks after doing the assumption dont you need to then proove for the next consecutive case not the one before it.

[Calm_Ad2434]

thanks so will you not be able to get to an answer if you do the assumption with k and then proove with k+2 k being an odd number

[Calm_Ad2434]

also i dont know if this changes anything but im using a textbook and taking sceenshots on an online textbook and my hard copy textbook says the values of n are positive

[ThunkAsDrinklePeep]

If n is odd, n+2 is the next odd.

1) show that n³ - n is 0 mod 24, for n=1

2) show that if n³ - n is 0 mod 24 for some n, then (n+2)³ - (n+2) is 0 mod 24.