Is it possible to prove... (1 Viewer)

[IAU001]

How do I prove:
(20²² - 17²² + 4³³ - 1) is divisible by 174.

One thing I've noticed is that if you divide all the exponents of the numbers by 11, which would become (20² - 17² + 4³ - 1) is exactly equal to 174. I don't know if that will help but that is one thing I noticed. Sorry I'm in Year 10 and I can't solve this question. Any help is greatly appreciated.

Thanks

 

[jb_nc]

Why wouldn't it be divisible? Why wouldn't any real number be able to divide that?

I'm not sure what you mean. This is probably more general related work.

 

[AMorris]

I'm guessing that this is from the Polya Enrichment Series? (sounds like one of those questions)

So with this question you have to use modular arithmetic and also notice the prime factorisation of 174 = 2*3*29

so basically the problem reduces to three sub problems of proving the expression is congruent to 0 mod 2; mod 3; mod 29. the mod 2 case is dead easy. the mod 3 case should pose any difficulty either. for the mod 29 case you have to take notice of the fact that the exponents are all multiples of 11 and use that fact to help you. If you are still making no progress ask again and I'll give some more hints.

 

[Zozo6969]

Mathematical induction

 

[IAU001]

Sorry AMorris but I'm only in Year 10, and as such don't have any clue as to what modular arithmetic is. Is there any other way to do this?

 

[kony]

modular arithmetic is a very simple concept (it was in year 7 maths enrichment) - so you definitely should be able to understand it - just look it up.

 

[jb_nc]

yeah theres a lesson on wikibooks you should be able to grasp it within 30mins-1hr.

 

[IAU001]

Oh lol woops...OK thanks I shall have a look.

 

[AMorris]

Mathematical induction

I don't see that as working because this is only really valid when the exponents are a specific set of numbers. These questions are very different to the 3u questions where you asked to prove the divisibility by induction and they require far stronger techniques.

And kagrawal - where did u get these questions from?

 

[watatank]

My friend is doing enrichment and he photocopied me the problems. I don't remember which stage he's doing but yeah. I can't do any of them and I top my year in Maths every year since year 7. That's how I got the problems. Also, if i can't do these problems, am I losing my touch? Because I'm aiming to do 3 unit next yr but this is making me double back and think about it.

no. ive gathered you are very good at maths, stick with three unit at the very least.

p.s. ive got no idea how to do that question (mostly cos i havent seen any of that kind and cbf looking it up) and i'm doing first year uni maths

 

[AMorris]

Don't worry about it all kagrawal. These sort of questions require vry different sort of thinking to those that you would get in 3u. And the mathematics required to solve this problem (modular arithmetic) is completely unneccessary for the 3u (or any nsw hsc) syllabus.

I don't think theres any other way of solving this problem using elementary mathematics without resorting modular arithmetic either.