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

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.

$\mathrm{Let}94\mathrm{be}\mathrm{the}n\mathrm{th}\mathrm{term}\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{AP}.\phantom{\rule{0ex}{0ex}}\therefore {a}_{\mathit{n}}=3+\left(n-1\right)\times 7\phantom{\rule{0ex}{0ex}}\Rightarrow 94=3+\left(n-1\right)\times 7\phantom{\rule{0ex}{0ex}}\Rightarrow 7\left(n-1\right)=94-3=91\phantom{\rule{0ex}{0ex}}\Rightarrow n-1=\frac{91}{7}=13\phantom{\rule{0ex}{0ex}}\Rightarrow n=14$

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

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