kms - fórum o príkladoch

6. príklad 1. letnej série 2016/2017

Zadanie:
New Yorčania volia, ktoré prirodzené číslo sa stane ich starostom. Čísla však nemajú svojich voliteľov, ale deliteľov. Nech $d_1,\, d_2,\, \dots,\, d_k$ spĺňajúce $1=d_1< d_2 < \dots < d_k=N$ sú všetky kladné delitele prirodzeného čísla $N\ge 2$ . Prirodzené číslo $N$ postúpi do druhého kola volieb práve vtedy, keď pre neho platí

$(d_1,d_2)+(d_2,d_3)+\dots+(d_{k-1},d_k)=N-2.$

Nájdite všetky prirodzené čísla $N$ , ktoré postúpia do druhého kola volieb. Nezabudnite zdôvodniť, že ste naozaj našli všetky čísla. \textit{Poznámka.} $(a,b)$ označuje najväčšieho spoločného deliteľa čísel $a,\,b$ .

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