Қалалық Жәутіков олимпиадасы 9 сынып, 2007 жыл

Мына түрдегі сандар
а) $\underbrace{11 \ldots 1}_{2007}2\underbrace{11 \ldots 1}_{2007}$;
б) $\underbrace{11 \ldots 1}_{2008}2\underbrace{11 \ldots 1}_{2008}$
жай сан бола ма?

Комментарий/решение

a) $A=\underset{2007}{\underbrace{11...1}}2\underset{2007}{\underbrace{11...1}}.$

$A=\underset{2008}{\underbrace{11...1}}\underset{2007}{\underbrace{00...0}}+\underset{2008}{\underbrace{11...1}}=\underset{2008}{\underbrace{11...1}}\cdot 10^{2007}+\underset{2008}{\underbrace{11...1}}=\underset{2008}{\underbrace{11...1}}\left ( 10^{2007}+1 \right ).$

б) $B=\underset{2009}{\underbrace{11...1}}\underset{2008}{\underbrace{00...0}}+\underset{2009}{\underbrace{11...1}}=\underset{2009}{\underbrace{11...1}}\cdot 10^{2008}+\underset{2009}{\underbrace{11...1}}=\underset{2009}{\underbrace{11...1}}\left ( 10^{2008}+1 \right ).$

Жай сан болмайды.