A *simple* maths problem.

[jack]

Q. Show that *=N⁴+N²+2N is a multiple of 4.
A. It's a multiple of N so changing N by four doesn't change the expression being a multiple of four or not[1]. So we only need to check N=0,1,2 and 3. But it works for those cases or you wouldn't have asked the question. QED.

Full credit? You would, I think, normally accept four one-line substitutions as sufficient? You don't actually write out the multiplications, you just say "1⁴+1²+2.1=4", and you could do the same in modular arithmetic and all the answers would be zero. And this is just a completely trivial collapse of that.

It's even realistic -- if you're proving S(n) for all n and find that S(n)∀n<N=>S(n)∀n [2] someone's probably crunched through N on a supercomputer already[3], and it each m must be S as if m wasn't they'd have published, the pleasure of a paper titled "Hahahahah!" and reading in its entirity "~S(m)" :)

But something about the answer bothers me. I think it's that (a) it's a smartarse and (b) a good habit to be into is to at least note down each case and put a tick by it to show you've tried them all in your head and didn't forget one.

[1] Yes, I could have used modular arithmetic language, but the colloqiual language fits the point better.
[2] Googling for "html math symbols" first produces a page suggesting you use the symbol font :(
[3] OK, professional problems are complicated enough that m might be too stupidly big. Witness the colouring theorem. But it's a major major major major[4] step, and someone else will hopefully fill in the blanks, even if they turn out to be hard.
[4] C22 reference.

Comments

[miss-next.livejournal.com]

No, there ought to be a more logical way to do it than that.

If N is even, then N^4, N^2 and 2N are automatically divisible by 4 (you can see that this is so by writing N = 2M, where M is an integer). Hence the result just pops out.

If N is odd, then you can write it as 2M + 1. Expanding the original expression gives you

(16M^4 + 16M^3 + 12M^2 + 4M + 1) + (4M^2 + 2M + 1) + 4M

which reduces to

16M^4 + 16M^3 + 16M^2 + 10M + 2

...

Ah. You need to prove that 10M + 2 is divisible by 4, but this follows only if M is odd, and I've almost run out of lunch hour.

Still, with a bit of luck this is a start!