$2^{2010} - 1 =2^0 + 2^1 + 2^2 + .....+ 2^{2009}$
Ta chứng minh $2^0 + 2^1 + 2^2 + .....+ 2^{2009} \vdots 31$
Đặt $A$ = $2^0 + 2^1 + 2^2 + .....+ 2^{2009}$
$⇒A=(2^0+2^1+2^2 + 2^3 + 2^4) + ....... + (2^{2005} + 2^{2006} + 2^{2007} + 2^{2008} +2^{2009})$
$⇔ A= 31 + ...... + 2^{2005}.31$
$⇔ A = 31. (1 + .... + 2^{2005})$$
$⇒ A \vdots 31$
$⇒$ $2^0 + 2^1 + 2^2 + .....+ 2^{2009} \vdots 31$
$⇒$ $2^{2010}-1 \vdots 31$
