How many natural numbers less than 1000 are divisible by 5 or 7 but NOT by 35? (a) 285 (b) 313 (c)341 (d) 243
i know that numbers upto 1000 that are divisible by 5 are 1000/5=200 and by 7 are 142. so total numbers that are divisible by 5 or 7 =342. then i am blank about how to progress. i guess we have to find area of divisibles of 5 and 7 excluding the area common between them in venn diagram representation.
Similarly, the number of numbers divisible by $35$ from $1$ to $999$ is $\left[\frac{999}{35}\right]=28$, so the answer is $199+142-2\times28=285$. Also, you are to use $999$ and not $1000$ (owing to the "less than $1000$" in your question) for getting your sort of Venn diagram areas. So, to put it exactly, it is
$$\left[\frac{999}{5}\right]+\left[\frac{999}{7}\right]-2\times\left[\frac{999}{35}\right]=285$$ Explanation to the Venn-diagram approach:
(i) There are $28$ numbers that are divisible by both $5$ and $35$(numbers divisible by $35$ are also divisible by $5$)
(ii) There are $28$ numbers that are divisible by both $7$ and $35$(numbers divisible by $35$ are also divisible by $7$)
These $28$ numbers each of which are being counted twice (once the first term and once in the second term) are to be removed from the enumeration.