Question : How many natural numbers less than 1000 are divisible by 5 or 7 but not by 35?
Option 1: 243
Option 2: 341
Option 3: 285
Option 4: 313
Correct Answer: 285
Solution :
The number of natural numbers less than 1000 that are divisible by 5.
$⇒\left\lfloor\frac{999}{5}\right\rfloor = 199$
The number of natural numbers less than 1000 that are divisible by 7.
$⇒\left\lfloor\frac{1000}{7}\right\rfloor = 142$
However, we've double-counted numbers that are divisible by both 5 and 7, i.e., numbers that are divisible by 35.
There are $\left\lfloor\frac{1000}{35}\right\rfloor = 28$ such numbers.
The total number of natural numbers less than 1000 that are divisible by 5 or 7 but not by 35 is $199+ 142 - 2\times28 = 285$
Hence, the correct answer is 285.
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