Inclusion/Exculsion Question

So I am doing some practice problems and I was wondering if someone can clarify what the question is asking.

The question:

a) How many integers between 1 and 400 (inclusive) are divisible by at least one of 4,5 and 20?

b) How many integers between 1 and 400 (inclusive) are divisible by just one of 4,5 and 20?

a) I am thinking the question is asking divisible by 4 and 5 and 20

b) I am thinking the question is asking divisible by one of 4 or 5 or 20

Can someone clarify for 'a' and 'b'?

Thanks.

Re: Inclusion/Exculsion Question

I think the question is pretty clear. Part a) talks about numbers n such that 4 divides n or 5 divides n or 20 divides n, i.e., any of the numbers 4, 5, 20 (or possible two or all three of them) divide n. Part b) talks about n such that there is exactly one number k in {4, 5, 20} such that k divides n.

Re: Inclusion/Exculsion Question

Quote:

Originally Posted by

**xboxwee** The question:

a) How many integers between 1 and 400 (inclusive) are divisible by at least one of 4,5 and 20?

b) How many integers between 1 and 400 (inclusive) are divisible by just one of 4,5 and 20?

Here is a bit more on a).

There are $\displaystyle \left\lfloor {\frac{{400}}{n}} \right\rfloor $ integers from 1 to 400 that are divisible by *n*.

Any integer that is divisible by $\displaystyle 20$ is also divisible by $\displaystyle 4~\&~5$. So in part a) we have to be careful not to count those numbers twice.

Re: Inclusion/Exculsion Question

Hello, xboxwee!

Quote:

a) How many integers between 1 and 400 (inclusive)

are divisible by **at least one** of 4, 5 and 20?

Every 4th number is divisible by 4:

. . $\displaystyle \frac{400}{4} \:=\:100$ numbers divisible by 4.

Every 5th number is divisible by 5:

. . $\displaystyle \frac{400}{5} \:=\:80$ numbers divisible by 5.

But every 20th number is divisible by 4 and 5:

. . $\displaystyle \frac{400}{20} \:=\:20$ numbers divisible by 20.

and these have been counted twice.

Therefore: .$\displaystyle 100 + 80 - 20 \:=\:160$

Quote:

b) How many integers between 1 and 400 (inclusive)

are divisible by **exactly one** of 4, 5 and 20?

Divisible by 4, but not 5 or 20: .$\displaystyle 100 - 20 \:=\:80$

Divisible by 5, but not 4 or 20: .$\displaystyle 80 - 20 \:=\:60$

Divisible by 20, but not 4 or 5: .impossible

Therefore: .$\displaystyle 80 + 60 \:=\:140$