From: Skouteris Dimitris (dimitris(@)impact.dyn.unipg.it)
Date: Δευ 03 Φεβ 2003 - 12:53:35 EET
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Kalhmera eis apantas kai apasas. Arketoi ta vrhkate ta yhfia kai euge.
Akolou0ei h apanthsh. As 3anadoume to provlhma :
> Estw o ari0mos pou se dekadikh morfh grafetai :
>
> 12345678910x109876543210y
>
> Poia einai ta yhfia x kai y wste o ari0mos na diaireitai me to 99 ;
Gia na diaireitai enas ari0mos me to 99 prepei kai arkei na diaireitai me to 9 kai to 11.
Ta krithria diairetothtas me tous duo autous ari0mous vasizontai sthn idia arxh : O ari0mos a^n - b^n diaireitai panta me ton a - b (opou ^ shmainei uywsh se dunamh). Fusika me a, b diaforetikous ari0mous.
Ara o 10^n - 1 diaireitai panta me to 9, enw o 10^n - (-1)^n diaireitai panta me to 11.
To krithrio diairetothtas me to 9 einai mallon pasignwsto : To a0roisma twn yhfiwn prepei na diaireitai me to 9. Edw to a0roisma einai 92+x+y, to opoio prepei na diaireitai me to 9. Afou oi x,y einai monoyhfioi, exoume h x+y=7 h x+y=16.
To krithrio diairetothtas me to 11 einai kapws ligotero gnwsto. Einai opws auto tou 9, mono pou anti na paroume to a0roisma twn yhfiwn, enallassoume a0roisma/diafora proxwrwntas apo yhfio se yhfio kai to apotelesma prepei na diaireitai me to 11. Edw to apotelesma einai 1-2+3-4+...+1-0+y=10-x+y pou prepei na diaireitai me to 11, dhladh y-x=1.
To mono zeugos (x,y) pou ikanopoiei tis sun0hkes einai to (3,4), opote to x=3 kai to y=4.
Swsta apanthsan (me seira apanthshs) oi :
Unknown Technologies
Irene Papaeconomou
Natalia Simeonidou
George Papargiris
Vasilis Solachidis
Vaggelis Kapoulas
Royal (Giwrgos Xaloulos)
Kala na pernate,
Dimitris
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