Problem Style Question - STUCK! (GCSE standard)

Sorry for an unhelpful thread name. Not sure what to call this.

Anyway, I've got a problem with my homework, and I'm really stuck on this question. It reads:

Consider the equation

an^2=182

where

a

is any number between 2 and 5 and

n

is a positive integer. What are the possible values of

n

?

Any help would be much appreciated. The answers are in the back of the book but I'm not sure how to get to them.

It would be easy if a was an integer between 2 and 5, but it says any number.

Thanks,
Bon

Reply 1

DFranklin

OK, well I'll assume you're stating the question correctly, even though it does sound quite weird.

Probably the most natural way of solving this is to just look at all the "remotely possible" values for n^2 and see which ones work.

e.g. 20^2 = 400, so an^2 = 182 gives a = 182/400. Since a >=2, this is impossible.

(That's a fairly useless example, but from what I've seen of your ability, I'm sure you can solve this from here, so I didn't want to give any more help).

Reply 2

bon

Thanks - I agree, it's a very odd question.

So are you suggesting there's no way of doing it apart from trial and improvement?

I was thinking you could help the trial and error by doing the following:

\\\displaystyle an^2=182\\ \\n^2=\frac{182}{a}\\ \\n=\frac{\sqrt{182}}{\sqrt{a}}\\ \\n=\frac{\sqrt{2\times 91}}{\sqrt{a}}

And then say that

\displaystyle \sqrt{91}\ \text {and}\ \sqrt{2}

must be factors of n? Or does that not work?

Reply 3

bon

Ok I realise how stupid that last line is :s:

Reply 4

DFranklin

bon

Ok I realise how stupid that last line is :s:

Good! Although to be fair, I think it's easy to make that kind of mistake when you have a question that mixes integers and reals.

But you can do better than simple trial and error: n^2 = 400 doesn't work because a ends up too small. Similarly n^2 = 1 doesn't work because a ends up too big. But given you know

2\le a \le 5

you should be able to find the possible range for n^2. Then just allow for the fact that n is an integer. (e.g. if you found that n^2 > 7, then you know