Question

How many numbers are there in the set S = {200, 201, 202, . . . , 800} which are divisible by neither of 5 nor 7?

• A. 411
• B. 412
• C. 410
• D. none of these

Answer 1

The correct option is A 411

Total numbers in the set = (800 - 200) + 1 = 601
Number of numbers which are divisible by 5
$=\frac{(800-200)}{5}+1=121$
Number of numbers which are divisible by 7
$=\frac{(798-203)}{7}+1=86$
Number of numbers which are divisible by both 5 & 7
$=\left(\frac{770-210}{35}\right)+1=17$
$\therefore$ Number of numbers which are either divisible by 5 or 7 or both
= (121 + 86) - 17 = 190
Thus the number of numbers in the given set which is neither divisible by 5 nor by 7 = 601 - 190 = 411.
Hence (a) is the correct option.

Alternatively,
Number of numbers less than 805 not divisible by 5 or 7 = 805 x $\frac{4}{5}\times \frac{6}{7}$ = 552
Number of numbers less than 210 not divisible by 5 or 7 = 210 x $\frac{4}{5}\times \frac{6}{7}$ = 144
Total numbers = 552 - 144 = 408
Hence, required number = 408 - 4 + 7 = 411