The Number Theory - 19/04/2013

See an answer or request an answer!

The Number Theory - 19/04/2013

Postby showmyiq » Fri Apr 19, 2013 6:55 am

Number Theory

If n is a positive and odd number, can you prove that:

n^8 – n^6 –n^4 +n^2 is dividable by 5760

n^2 defines power 2 of n
n^2=n*n
n^3=n*n*n
etc.
User avatar
showmyiq
Site Admin
 
Posts: 390
Joined: Sat Sep 15, 2012 9:45 pm

Re: The Number Theory - 19/04/2013

Postby DArk0n3 » Sat Apr 20, 2013 6:02 pm

We first start by doing some rearrangements in order to make the polynom easier to work with.

n^8 - n^6 - n^4 + n^2 = n^2(n^6 - n^4 - n^2 +1) = n^2(n^4(n^2 - 1) - (n^2 - 1)) = n^2(n^2 - 1)(n^4 - 1) =
n^2(n - 1)(n +1)(n - 1)(n+1)(n^2 + 1)
Now if n is an odd number this means that n - 1 is an even number and n + 1 is an even number as well. And since n - 1 and n + 1 are two consecutive even numbers this means that one of them is divisible by 4. Then the polynom (n - 1)(n + 1) is divisible by 8. So the whole polynom n^2(n - 1)(n +1)(n - 1)(n+1)(n^2 + 1) is divisible by 64. Since n is odd this means that n^2 is also odd and n^2 + 1 is even. So n^2 + 1 is divisible by 2. So the whole polynom is n^2(n - 1)(n +1)(n - 1)(n+1)(n^2 + 1) is divisible by 128. The numbers n - 1; n; n + 1 are three consecutive numbers, so one of them is divisible by 3. This means that n(n - 1)(n + 1) is divisible by 3 and n^2(n - 1)(n + 1)(n - 1)(n +1) is divisible by 9. So our polynom is divisible by 128*9 = 1152. Now the last part - we have to prove that our polynom is divisible by 5. If n is divisible by 5 then we have no problem - the polynom is divisible by 5. If n is not divisible by 5 then it will be either of the kind 5m + 2 or of the kind 5m + 4, because it is an odd number. If it is of the kind 5m + 2 then n^2 + 1 = (5m + 2)^2 + 1 = 25m^2 +20m + 4 + 1 = 25m^2 + 20m + 5 = 5(5m^2 + 4m + 1), which is divisible by 5. If n is of the kind 5m + 4 then n + 1 will be equal to 5m + 5, which is divisible by 5. So in all cases our polynom is divisible by 5. This means that n^2(n - 1)(n +1)(n - 1)(n+1)(n^2 + 1) is divisible by 1152*5 = 5760
DArk0n3
 
Posts: 76
Joined: Sat Oct 06, 2012 4:20 pm

Re: The Number Theory - 19/04/2013

Postby showmyiq » Sat Apr 20, 2013 6:17 pm

Correct logic and my approach of solving this task was the same.

One interesting think – I have found this task in one Russian textbook. There was no solution about it, except that I should use induction in order to solve it. I try to attack the task via many different induction methods (without using the technics of dividable numbers) – but with no luck. I am pretty sure induction is not the right approach when solving such math tasks. What is your opinion on that matter? Can we attack the very same problem using induction?
User avatar
showmyiq
Site Admin
 
Posts: 390
Joined: Sat Sep 15, 2012 9:45 pm

Re: The Number Theory - 19/04/2013

Postby DArk0n3 » Sat Apr 20, 2013 6:26 pm

I tried using induction as well, but I couldn't find a proof. I don't think induction is the right method to approach this problem.
DArk0n3
 
Posts: 76
Joined: Sat Oct 06, 2012 4:20 pm

Re: The Number Theory - 19/04/2013

Postby showmyiq » Sat Apr 20, 2013 6:41 pm

Good!
Your name will be published as the one who solved the puzzle.
Tomorrow I will publish a math problem requiring induction in order to be solved. Stay tuned! :)
User avatar
showmyiq
Site Admin
 
Posts: 390
Joined: Sat Sep 15, 2012 9:45 pm

Re: The Number Theory - 19/04/2013

Postby DArk0n3 » Sat Apr 20, 2013 6:42 pm

Can't wait! :)
DArk0n3
 
Posts: 76
Joined: Sat Oct 06, 2012 4:20 pm


Return to Answers

Who is online

Users browsing this forum: No registered users and 1 guest

cron