Results 1 to 4 of 4

Math Help - Inclusion/Exculsion Question

  1. #1
    Newbie
    Joined
    Dec 2011
    Posts
    5

    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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,536
    Thanks
    778

    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.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,649
    Thanks
    1597
    Awards
    1

    Re: Inclusion/Exculsion Question

    Quote Originally Posted by xboxwee View Post
    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 \left\lfloor {\frac{{400}}{n}} \right\rfloor integers from 1 to 400 that are divisible by n.
    Any integer that is divisible by 20 is also divisible by 4~\&~5. So in part a) we have to be careful not to count those numbers twice.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,738
    Thanks
    644

    Re: Inclusion/Exculsion Question

    Hello, xboxwee!

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

    Every 5th number is divisible by 5:
    . . \frac{400}{5} \:=\:80 numbers divisible by 5.

    But every 20th number is divisible by 4 and 5:
    . . \frac{400}{20} \:=\:20 numbers divisible by 20.
    and these have been counted twice.

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




    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: . 100 - 20 \:=\:80

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

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

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

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Inclusion and exclusion question
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: April 19th 2011, 04:00 AM
  2. Question about a Question: Inclusion-Exclusion theorm
    Posted in the Advanced Statistics Forum
    Replies: 5
    Last Post: January 27th 2011, 12:51 PM
  3. inclusion & exclusion question help
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: May 17th 2010, 07:49 AM
  4. inclusion-exclusion question
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: May 16th 2010, 11:44 PM
  5. inclusion/exclusion question
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: November 3rd 2009, 07:31 PM

Search Tags


/mathhelpforum @mathhelpforum