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

Explanation:-

$$\eqalign{ & {\text{The required number}} \cr & {\text{ = Number of numbers, which are}} \cr & {\text{(divisible by 3 + divisible by 7 - divisible by 21)}} \cr & {\text{Number}}\,{\text{of}}\,{\text{number}}\,{\text{divisible}}\,{\text{by}}\,{\text{3,}} \cr & {\text{ = }}\left[ {\frac{{\left( {198 - 3} \right)}}{3}} \right] + 1 \cr & = 66 \cr & {\text{Number}}\,{\text{of}}\,{\text{number}}\,{\text{divisible}}\,{\text{by}}\,7, \cr & = \left[ {\frac{{\left( {196 - 7} \right)}}{7}} \right] + 1 \cr & = 28 \cr & {\text{Number}}\,{\text{of}}\,{\text{number}}\,{\text{divisible}}\,{\text{by}}\,21, \cr & = \left[ {\frac{{\left( {189 - 21} \right)}}{{21}}} \right] + 1 \cr & = 9 \cr & {\text{Thus,}}\,{\text{the}}\,{\text{divisible}}\,{\text{value}} \cr & = \left( {66 + 28 - 9} \right) \cr & = 85 \cr & {\text{Thus, number of numbers which are not divisible by }}3{\text{ or }}7 \cr & = {\text{ }}200{\text{ }} - {\text{ }}85 \cr & = {\text{ }}115 \cr}$$