Solve this:
Q). Find the remainder when ${\left(106\right)}^{85}-{\left(85\right)}^{106}+1$ is divisible by 7.

Dear Student,
Please find below the solution to the asked query:

$$\begin{gathered} {\left(106\right)}^{85}-{\left(85\right)}^{106}+1 \\ ={\left(15\times 7-1\right)}^{85}-{\left(7\times 12+1\right)}^{106}+1 \\ \mathrm{In} \mathrm{expansion} \mathrm{of} \mathrm{both} {\left(15\times 7-1\right)}^{85} \mathrm{and} {\left(7\times 12+1\right)}^{106}. \mathrm{all} \mathrm{terms} \mathrm{except} \\ \mathrm{last} \mathrm{terms} \mathrm{will} \mathrm{be} \mathrm{mutiple} \mathrm{of} 7 \\ \Rightarrow {\left(15\times 7-1\right)}^{85}-{\left(7\times 12+1\right)}^{106}+1 \\ =7{\mathrm{k}}_1{+}^{85}{\mathrm{C}}_{85}{\left(-1\right)}^{85}-\left[7{\mathrm{k}}_2{+}^{106}{\mathrm{C}}_{106} 1^{106}\right]+1 \\ =7{\mathrm{k}}_1-1-7{\mathrm{k}}_2-1+1 \\ =7\left({\mathrm{k}}_1-{\mathrm{k}}_2\right)-1 \\ =7\left({\mathrm{k}}_1-{\mathrm{k}}_2\right)-7+6 \\ =7\left({\mathrm{k}}_1-{\mathrm{k}}_2-1\right)+6 \\ \mathrm{Hence} \mathrm{remainder} \mathrm{is} 6 \end{gathered}$$

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