The numbers ${\mathrm{a}}_{\mathrm{n}}=6^{\mathrm{n}}-5\mathrm{n}$ for n = 1,2,3, ....... when divided by 25 leave the remainder

$\begin{array}{rcl}{\mathrm{a}}_{\mathrm{n}}& =& 6^{\mathrm{n}}-5\mathrm{n}\\ & =& {(1+5)}^{\mathrm{n}}-5\mathrm{n} \\ & =& 1+5\mathrm{n}+\mathrm{n}{5}^2+{}^{\mathrm{n}}{\mathrm{c}}_3 5^3+.......+{}^{\mathrm{n}}{\mathrm{c}}_{\mathrm{n}} 5^{\mathrm{n}}-5\mathrm{n}\\ & =& 1+5^2 \left({}^{\mathrm{n}}{\mathrm{c}}_2+{}^{\mathrm{n}}{\mathrm{c}}_3 5+.....+ {}^{\mathrm{n}}{\mathrm{c}}_{\mathrm{n}} 5^{\mathrm{n}-2}\right)\\ & =& 1+25\mathrm{k}, \mathrm{k}\in \mathrm{Z}\end{array}$

${\mathrm{a}}_{\mathrm{n}}$ divided by 25 the remainder is 1