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1. The number of numbers from 1 to 200 which are divisible by neither 3 nor 7 is :

A. 115

B. 106

C. 103

D. less than 100

Solution:

The required number = Number of numbers, which are (divisible by 3 + divisible by 7 - divisible by 21)
Number of number divisible by 3,

\begin{aligned} = \frac{198 - 3}{3} + 1 \\ = 66 \end{aligned}

Number of number divisible by 7

\begin{aligned} = \frac{196 - 7}{7} + 1 \\ = 28 \end{aligned}

Number of number divisible by 21,

\begin{aligned} = \frac{189 - 21}{21} + 1 \\ = 9 \end{aligned}

Thus, the divisible value

\begin{aligned} = 66 + 28 - 9 \\ = 85 \end{aligned}

Thus, number of numbers which are not divisible by 3 or 7

\begin{aligned} = 200 - 85 \\ = 115 \end{aligned}

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