a) Show that n(2n + 1)(7n + 1) is divisible by 6 for all integers n.

b) Find all integers n such that n(2n + 1)(7n + 1) is divisible by 12.



So far here is what I have:  For part a), I found that either n or 7n+1 is even.  Using congruences and modular arithmetic, I also found that no matter what congruence n has to 3, one of the three of n, 2n+1, or 7n+1 are divisible by three, so I solved part a).

Part b) is the part I am stuck on.  I know that exactly one of n and 7n+1 is always even for any integer n, and I need to find integers n that one of n or 7n+1 is divisible by 4.  Could someone give me a hint? Thanks in advance!

 Aug 6, 2020

Since you already proved that n(2n + 1)(7n + 1) is divisible by 3, now you need to see how it could be divisible by 4. We have already seen that 2n+1 is odd, so n(7n+1) has no be divisible by 4. This happens when n is divisible by 4, or when 7n+1 is divisible by 4. Now, we first need to see when 7n+1 is divisible by 4. As you see, this happens if and only if -n-1 is divisible by 4. (by subtracting 8n) From there, can you generalize an equation that is true for n? (In the problem, you do not have to list out e v e r y   s i n g l e option, as there are infinite possibilities.)


(please ask for more help if needed!)



 Aug 6, 2020

Alright, thanks!  I will try what you said :)

 Aug 6, 2020

Have you gotten the answer, or do you need more help?

ilorty  Aug 6, 2020

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