A *simple* maths problem.
Apr. 26th, 2006 12:20 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
jackQ. 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.
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.