If each of the two numbers 5¹⁶ and 5²⁵ are divided by 6, the remainders are R₁ and R₂ respectively. What is the value of $$\frac{{{{\text{R}}_1} + {{\text{R}}_2}}}{{{{\text{R}}_2}}}?$$

Solution (By Examveda Team)

Given:
Two numbers 5¹⁶ and 5²⁵ are divided by 6, the remainders are R₁ and R₂ respectively.
Concept used:
If a number in the form of (b - 1)ⁿ is divided by b
Then,
Remainder = 1 if n is even number
Remainder = (b - 1) if n is an odd number
Calculation:
$$\eqalign{ & {5^{16}} = {\left( {6 - 1} \right)^{16}} \cr & {\text{So, remainder}} = \frac{{{{\left( {6 - 1} \right)}^{16}}}}{6} = 1\,{\text{i}}{\text{.e}}{\text{.,}}\,{{\text{R}}_{\text{1}}} \cr & {\text{Again, }}{5^{25}} = {\left( {6 - 1} \right)^{25}} \cr & {\text{So, remainder}} = \frac{{{{\left( {6 - 1} \right)}^{25}}}}{6} = \left( {6 - 1} \right) = 5\,{\text{i}}{\text{.e}}{\text{.,}}\,{{\text{R}}_{\text{2}}} \cr & {\text{Now,}} \cr & \frac{{{{\text{R}}_1} + {{\text{R}}_2}}}{{{{\text{R}}_2}}} = \frac{{1 + 5}}{5} = \frac{6}{5} \cr & \therefore {\text{Required answer is }}\frac{6}{5} \cr} $$