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The Mathematical olympiad forum?
Topic Started: Mar 31 2009, 10:01 AM (1,064 Views)
Einstein the 2nd
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Firstly, I do not think that anyone will come here.
Secondly, I am asking help because i really do not know how to do these questions and i am not ashamed to ask questions because, if you never ask you'll never know. ( Hitler don't sprout crap about this thing, i know its wrong)
Thirdly,STRICTLY NO DIGRESSING IN THIS FORUM LIKE YOU DO FOR THE OTHERS. i.e. from math olympiad to syazani being short, to "Yes you finally get it!", etc.

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Einstein the 2nd
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Ok, so erm... how do you do this

Prove that for every integer n, the number 77n +1 is the product of at least 2n + 3 primes. (the primes do not have to be distinct)

Solution:
let A(n) = 77n +1. So lets assume its as the question says and has 2n+3 prime divisors.

Then we can express A(n+1) as [(A(n)-1)7] +1. Go do your binomial theorem, and you will know why. there is symetry right? 1,7,21,35,35,21,7,1.

the there's a (A(n)6) and the rest...which after factoring, we get:
(A(n+1) ) = A(n)(A(n))6 - 7 (A(n)-1)( A(n))2 - A(n) + 1)2)

so, from this, 7(A(n)-1) = 77n+1 = 72k

The 2k just expresses the thingy i said earlier on which also makes the number a square.

Perfect squares. So the thingy i said earlier on is also a perfect square!

Then with difference of squares property, It is a product of 2 numbers, so + 2 prime factors!
so A(n+1)= 2 + 2n + 3 prime factors, hence, proven.

QED!!! YAYAYAY!!

[/sup][/sup]
Edited by Einstein the 2nd, Apr 4 2009, 12:39 PM.
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Williamcxp
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and no one replies..
not even hitler..
sorry i do not understand the question
much less know how to do..
but i would be interested to be taught. ^^
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Hitler
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i am not helping you with this unless you help me with the modulo part. i have not tried yet but i already know you specially chose a question requiring the use of it to tick me off.
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Hitler
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k last post of the day bye now...
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Einstein the 2nd
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I can't even solve it??
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Hitler
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you do not have to know it requires modulo. you just have to do it until you get stuck at the modulo part.
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Einstein the 2nd
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fine, thanks for reminding me.... ahaha just joking.
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Hitler
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of course you were. if you were not you would not be able to post that reply.
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Einstein the 2nd
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Stupid.
Edited by Einstein the 2nd, Apr 1 2009, 01:02 PM.
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