How Many Positive Integers Below $201$ Are Multiples of $4$ But Not Multiples of $10$?

Question

I have recently been starting to work on counting problems, and currently, I am working on this one, but cannot figure it out:

How many positive integers less than 201 are multiples of 4 but not multiples of 10?

I was also wondering if there was a specific formula for this type of problem, so I can fully understand it in the future. Any help would be appreciated, and thanks in advance

Answer 1

Well Let $A = $ the number of positive multiples of $4$ that are less than $201$.

One out of ever $4$ is a multiple of $4$ there are $200$ positive integers less than $201$ (they are $1,2,3,......, 200$ so there are $\frac {200}{4} = 50$ that are multiples of $4$. $A = 50$.

Suppose $n$ is both a multiple of $4$ and of $10$ what can we say about $n$?

If an number is a multiple of $4$ and $10$ then it is a multiple of $\operatorname{lcm}(4,10)=\operatorname{lcm}(2^2, 2\times 5)=2^2\times 5=20$.

Let $B$ be the number of integers that can be the thing in the hint above, that is a multiple of both $4$ and $10$.

Every $20$ numbers is a multiple of $20$. so there are $\frac {200}{20} = 10$ multiples of $20$. So $B=10$.

Then the number of multiples of $4$ that aren't multiples of $10$ will be all the mulitples of $4$ less the multiples of $4$ that are multipls of $10$ so that number is $A-B$.

So your answer is $A-B$.

$A-B = 50 -10 = 40$.

Answer 2

You could split this into two successive questions:

1. How many positive integers below $201$ are multiples of $4$?
2. How many of those numbers are multiples of $10$?

The first part is fairly easy - they are the numbers $\{4,8,12,16,20,24,\ldots, 196,200\}$.

Now also eliminating those divisible by $10$ is straightforward; the set to remove will look like $\{20,\ldots,200\}$

Hopefully you can work out the size of both these sets to answer the question.

With a little thought, this should also give you a method of calculating the answer in other questions of this sort. It's worth double-checking the end-points of the included/excluded numbers.