number of natural number less than or equal to 1000 which is not divisible by 10 , 15 or 25

Question

This is a strange question for me. I would be glad if someone here provide me with an appropirate answer and the procedure of solving it.

Answer 1

The number of numbers divisible by $10$ is $[\frac{1000}{10}]=100$.

The number of numbers divisible by $15$ is $[\frac{1000}{15}]=66$.

The number of numbers divisible by $25$ is $[\frac{1000}{25}]=40$.

The number of numbers divisible by both $10$ and $15$ is $[\frac{1000}{30}]=33$.

The number of numbers divisible by both $10$ and $25$ is $[\frac{1000}{50}]=20$.

The number of numbers divisible by both $15$ and $25$ is $[\frac{1000}{75}]=13$.

The number of numbers divisible by $10$, $15$ and $25$ is $[\frac{1000}{150}]=6$.

Hence there are $100+66+40-33-20-13+6=146$ which are divisible by one of your numbers. Finally, there are $1000-146=854$ numbers which are not.

A little more insight:

The idea is that the first $3$ numbers $(100,66,50)$ represent how many integers in your range are divisible by the respective integers. However, some of them are counted twice, that's why we need to see how many of them appear in $2$ of the respective piles of numbers - for example, all numbers divisible by both $10$ and $15$ will appear in both the first and the second pile. Thus, after we find the number of such integers, we have to substract it. Finally, there are numbers that have appeared $3$ times initially and have been substracted $3$ times - hence we need to add them to the list. Those are precisely the numbers that are divisible by all of $10,15$ and $25$.

If you are more interested, have a look at the Inclusion-Exclusion Formula.

Answer 2

That set is the same as the set of all natural numbers $\leq 1000$ minus those divisible by 10, 15 and 25.

The numbers divisible by 10, 15 and 25 are the same than those divisible by 150 (It is pretty much the property of the least common multiple). There are $\left \lfloor{\frac{1000}{150}}\right \rfloor=6 $ such numbers.

Therefore, there are 1000-6=994 numbers $\leq 1000$ that are not divisible by 10, 15 or 25.