[tex]S = 2^{26}+2^{27}+...+2^{53} \\ 2S = \quad \,\,\,\,\,\,\,2^{27}+...+2^{53}+2^{54} \\ \noindent\rule{5cm}{0.7pt} \\ 2S-S = 2^{54}-2^{26} \\ S = 2^{54}-2^{26}\\ S = 4^{27}-4^{13} \\S = 4^{2\cdot 13+1}-4^{2\cdot 6+1}\\ S = 16^{13}\cdot 4 - 16^{6}\cdot 4 \\ \\ \text{Aplic formula: }(a+b)^n = M_a+b^n \\ \\ S = (15+1)^{13}\cdot 4 - (15+1)^{6}\cdot 4 \\ S = (M_{15}+1^{13})\cdot 4 - (M_{15}+1^6)\cdot 4 \\ S = 4\cdot (M_{15}+1-M_{15}-1) \\ S = 4\cdot M_{15} \\ \\\Rightarrow S = M_{15} \Rightarrow S \, \vdots\, 15[/tex]
