Question

find the number of three digit natural numbers which are divisible by 11.

Answer 1

$$\begin{gathered} \mathrm{The}\mathrm{list}\mathrm{of}3-\mathrm{digits}\mathrm{natural}\mathrm{numbers}\mathrm{divisible}\mathrm{by}11\mathrm{are}: \\ 110,121,132,...............990 \\ \mathrm{The}\mathrm{above}\mathrm{list}\mathrm{is}\mathrm{in}\mathrm{AP}\mathrm{with}\mathrm{first}\mathrm{term}, \\ a\mathit{}=110\mathrm{and}\mathrm{common}\mathrm{difference},\mathit{}d=11 \\ \mathrm{Let}n\mathit{}=\mathrm{number}\mathrm{of}\mathrm{terms} \\ \mathrm{then}\mathit{}{a}_n=990 \\ \Rightarrow a+\left(n-1\right)d=990 \\ \Rightarrow 110+\left(n\mathit{}-1\right)\times 11=990 \\ \Rightarrow 110+11n-11=990 \\ \Rightarrow 99+11n=990 \\ \Rightarrow 11n=990-99 \\ \Rightarrow 11n=891 \\ \Rightarrow \mathit{}n=\frac{891}{11}=81 \\ \mathbf{So}\mathbf{,}\mathbf{}\mathbf{th}\mathbf{e}\mathbf{}\mathbf{3}\mathbf{-}\mathbf{digits}\mathbf{}\mathbf{natural}\mathbf{}\mathbf{numbers}\mathbf{}\mathbf{which}\mathbf{}\mathbf{are}\mathbf{}\mathbf{divisible}\mathbf{}\mathbf{by}\mathbf{}\mathbf{11}\mathbf{}\mathbf{are}\mathbf{}\mathbf{81}\mathbf{.} \end{gathered}$$