how many positive integers less than 100 have a remainder of 3 when divided by 7

Dear Student,

Here is the answer to your query.

The positive integers less than 100 that leaves remainder 3 when divided by 7 are 3, 10, 17, ..., 94.
3, 10, 17, ..., 94 are in AP, whose first term, a = 3 and common difference, d =7.

$$\begin{gathered} \mathrm{Let} 94 \mathrm{be} \mathrm{the} n\mathrm{th} \mathrm{term} \mathrm{of} \mathrm{the} \mathrm{given} \mathrm{AP}. \\ \therefore a_{\mathit{n}}=3+\left(n-1\right)\times 7 \\ \Rightarrow 94=3+\left(n-1\right)\times 7 \\ \Rightarrow 7\left(n-1\right)=94-3=91 \\ \Rightarrow n-1=\frac{91}{7}=13 \\ \Rightarrow n=14 \end{gathered}$$

Thus, there are 14 positive integers less than 100 which have a remainder 3 when divided by 7.