How many numbers are there from 700 to 950 (including both) which are neither divisible by 3 nor by 7?

Solution(By Examveda Team)

Number which is divisible by 3
$$\eqalign{ & \frac{{700}}{3} - \frac{{950}}{3} \cr & = 233 - 316 \cr & = 83......\left( {\text{i}} \right) \cr} $$
⇒ Number which is divisible by 7
$$\eqalign{ & \frac{{700}}{7} - \frac{{950}}{7} \cr & = 100 - 135 \cr & = 35 + 1 \cr & = 36......\left( {{\text{ii}}} \right) \cr} $$
(∴ If 1^st number is totally divisible by 7 then we add +1 in final result)
⇒ Number which is divisible by 21 (LCM of 3 or 7)
$$\eqalign{ & \frac{{700}}{{21}} - \frac{{950}}{{21}} \cr & = 33 - 45 \cr & = 12......\left( {{\text{iii}}} \right) \cr} $$
∴ Number which is divisible by 3 or 7
(i) + (ii) - (iii)
(83 + 36) - 12 = 119 - 12 = 107
Total number from 700 to 950
n = (950 - 700) + 1 = 251
∴ Number which is neither divisible by 3 nor 7
⇒ 251 - 107 = 144