3 Steps To Find Out The Number Of Natural Numbers That Are Divisible By Certain Numbers

“How many natural numbers between 200 and 500 are divisible by 2,3,4,5 and 6?”

This is an interesting question which I have found online when I did a Google search for maths olympiad questions. What I found particularly interesting is that you can approach this question either using ‘guess-and-check’ or an understanding of the properties of numbers. Of course, these are not the only ways to solve this question and our readers may have more approaches to this sum.

In this article, I will demonstrate an approach to this question based on the understanding of the divisibility of the numbers of 2,3,4,5 and 6. I find this approach a very neat way of solving this question and students can solve this question in less than 2 minutes.

But first, what are natural numbers?

What are natural numbers?

According to the Wolfram website, natural numbers refers to a set of positive integers such as 1,2,3… or to a set of non-negative integers 0,1,2,3… or simply put, they are numbers used for counting. There is also a debate of whether to include the number 0 under the category of natural numbers but that is not the topic of this article.


So how do you approach this question?

You can try and divide the range of numbers amongst 2,3,4,5 and 6 but this approach would be too tedious.

Step 1: Simplify the problem through observation

A simple method might be to observe that the number 2 is a factor of both 4 (2 x2) and 6(3×2). If the numbers can be divided by 2, they can also be divided by 4 and 6,thus we only need to focus on numbers 2,2,3 and 5.

Step 2:Multiply the numbers

Multiplying the numbers, 2x2x3x5 gives us 60. So if any number within the range of 200 to 500 can be divided by 60, then that is the natural number we are looking for.

Step 3: Conclusion

So the natural numbers are 240, 300,360, 420 and 480. A total of 5 natural numbers.

I hope this approach is fast and simple way to quickly find out the number of natural numbers that are divisible by 2,3,4,5 and 6.

Using the above example, try to generalise the approach to similar questions.

Source by Penny Chow