Question

How many numbers from 0 to 999 are not divisible by either 5 or 7 ? (a) 313 (b) 341 (c) 686 (d) 786

Answer 1

$Find\ no.\ of\ numbers\ from\ 0\ to\ 999\ which\ are\ divisible\ by\ 35. \\ 0\ will\ be\ includ\[ed. \\ \\ 0\ is\ the\ smallest\ multiple\ of\ 35. \\ \\ 999\ divided\ by\ 35\ gives\ quotient\ 28. \\ \\ \therefore\ 28 \times 35 = 980\ is\ the\ largest\ multiple\ of\ 35.$

$Consider\ multiples\ of\ 35\ are\ in\ an\ AP. \\ \\ T_1 = 0 \\ \\ T_n = 980 \\ \\ d = 35 \\ \\ n = \frac{T_n - T_1}{d} + 1 \\ \\ = \frac{980 - 0}{35} + 1 = \frac{980}{35} + 1 = 28 + 1 = 29 \\ \\ \therefore\ There\ are\ 29\ multiples\ of\ 35\ from\ 0\ to\ 999.$

$Find\ no.\ of\ numbers\ from\ 0\ to\ 999\ which\ are\ divisible\ by\ 5. \\ 0\ will\ be\ includ\[ed. \\ \\ 0\ is\ the\ smallest\ multiple\ of\ 5. \\ \\ 999\ divided\ by\ 5\ gives\ quotient\ 199. \\ \\ \therefore\ 199 \times 5 = 995\ is\ the\ largest\ multiple\ of\ 5.$

$Consider\ multiples\ of\ 5\ are\ in\ an\ AP. \\ \\ T_1 = 0 \\ \\ T_n = 995 \\ \\ d = 5 \\ \\ n = \frac{T_n - T_1}{d} + 1 \\ \\ = \frac{995 - 0}{5} + 1 = \frac{995}{5} + 1 = 199 + 1 = 200 \\ \\ \therefore\ There\ are\ 200\ multiples\ of\ 5\ from\ 0\ to\ 999.$

$Find\ no.\ of\ numbers\ from\ 0\ to\ 999\ which\ are\ divisible\ by\ 7. \\ 0\ will\ be\ includ\[ed. \\ \\ 0\ is\ the\ smallest\ multiple\ of\ 7. \\ \\ 999\ divided\ by\ 7\ gives\ quotient\ 142. \\ \\ \therefore\ 142 \times 7 = 994\ is\ the\ largest\ multiple\ of\ 7.$

$Consider\ multiples\ of\ 7\ are\ in\ an\ AP. \\ \\ T_1 = 0 \\ \\ T_n = 994 \\ \\ d = 7 \\ \\ n = \frac{T_n - T_1}{d} + 1 \\ \\ = \frac{994 - 0}{7} + 1 = \frac{994}{7} + 1 = 142 + 1 = 143 \\ \\ \therefore\ There\ are\ 143\ multiples\ of\ 7\ from\ 0\ to\ 999.$

$\\ \\ \\ Add\ both\ the\ no.\ of\ multiples\ of\ 5\ and\ 7\ and\ then\ subtract\ no.\ of \\ multiples\ of\ 35\ from\ it. \\ \\ 200 + 143 - 29 = 314 \\ \\ \therefore\ There\ are\ 314\ numbers\ from\ 0\ to\ 999\ which\ are\ divisible\ by \\ either\ 5\ or\ 7.$

$Totally,\ there\ are\ 999 - 0 + 1 = 1000\ numbers\ from\ 0\ to\ 999. \\ \\ Subtract\ 314\ from\ 1000. \\ \\ 1000 - 314 = \bold{686} \\ \\ \therefore\ There\ are\ \bold{686}\ numbers\ from\ 0\ to\ 999\ which\ are\ divisible\ by \\ neither\ 5\ nor\ 7. \\ \\ \therefore\ Option\ (c)\ is\ correct.$

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