The remainder of $$\frac{{{2^{59}}}}{{255}}$$ is:

$$\eqalign{ & {2^{59}}\,{\text{can}}\,{\text{be}}\,{\text{expressed}}\,{\text{as}}\,{2^{8n}} \cr & {\text{So}}, \cr & {\text{Remainder}}\,\frac{{{2^{59}}}}{{255}} \cr & = {\text{Remainder}}\,\frac{{{2^{8n}}}}{{255}} \cr & = {\text{Remainder}}\,\frac{{{{256}^n}}}{{255}} = 1 \cr & {\text{Required}}\,{\text{remainder}} = 1 \cr} $$