Question

plz ans........................​

Answer 1

$\text{First we have to expand \ 15^{23}+23^{23} } \\ \\ \\ \text{When \ 15^{23}+23^{23} \ \ is expanded, it seems like,} \\ \\ \\ \Longrightarrow\ 15^{23}+23^{23} \\ \\ \Longrightarrow\ (15+23)(15^{22}-15^{21} \cdot 23+15^{20}\cdot 23^2-15^{19}\cdot 23^3+......+23^{22}) \\ \\ \Longrightarrow\ 38(15^{22}-15^{21} \cdot 23+15^{20}\cdot 23^2-15^{19}\cdot 23^3+......+23^{22}) \\ \\ \Longrightarrow\ 19 \times 2(15^{22}-15^{21} \cdot 23+15^{20}\cdot 23^2-15^{19}\cdot 23^3+......+23^{22})$

$\text{At the last line, it seems that \ 15^{23}+23^{23} \ \ is divisible by 19.} \\ \\ \\ \large \underline{\underline{\text{Thus, the remainder on dividing 15^{23}+23^{23} by 19 is}\ \textbf{0}.}} \\ \\ \\ \normalsize \text{And the quotient is \ 2(15^{22}-15^{21} \cdot 23+15^{20}\cdot 23^2-15^{19}\cdot 23^3+......+23^{22}) }$