Results 1 to 3 of 3

Math Help - inclusion/exclusion

  1. #1
    Member
    Joined
    Oct 2008
    Posts
    124

    inclusion/exclusion

    Use the principle of inclusion/exclusion to determine how many numbers between 1 and 1000 are
    a) not divisible by 2, 3, 5, or 7

    b) not perfect n-th powers for any n>1 (not perfect squares, not perfect cubes, etc)

    c) not square free (not divisible by n^2 for any n>1)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,686
    Thanks
    616
    Hello, noles2188!

    Here's the first one . . .


    Use the principle of inclusion/exclusion to determine how many numbers
    between 1 and 1000 are: .a) not divisible by 2, 3, 5, or 7
    We will find how many numbers are divisible by 2, 3, 5, or 7.


    . . \begin{array}{c|c}<br />
\text{Div.by} & \text{Quantity} \\ \hline \\[-4mm]<br /> <br />
2 & \left[\tfrac{1000}{2}\right] \:=\:500 \\ \\[-3mm]<br />
3 & \left[\tfrac{1000}{3}\right] \:=\:333 \\ \\[-3mm]<br />
5 & \left[\frac{1000}{5}\right] \:=\:200 \\ \\[-3mm]<br />
7 & \left[\frac{1000}{7}\right] \:=\:142  <br />
\end{array} \qquad \text{Total: }1175


    . . \begin{array}{c|c}<br />
\text{Div.by} & \text{Quantity} \\ \hline \\[-3mm]<br /> <br />
2,3 & \left[\frac{1000}{6}\right] \:=\:166 \\ \\[-3mm]<br />
2,5 & \left[\frac{1000}{10}\right] \:=\:100 \\ \\[-3mm]<br />
2,7 & \left[\frac{1000}{14}\right] \:=\:71 \\ \\[-3mm]<br />
3,5 & \left[\frac{1000}{15}\right] \:=\:66 \\ \\ [-3mm]<br />
3,7 & \left[\frac{1000}{21}\right] \:=\:47 \\ \\[-3mm]<br />
5,7 & \left[\frac{1000}{35}\right] \:=\:28 \end{array}\qquad\text{Total: }414


    . . \begin{array}{c|c}<br />
\text{Div.by} & \text{Quantity} \\ \hline \\[-3mm]<br />
2,3,5 & \left[\frac{1000}{30}\right] \:=\:33 \\ \\[-3mm]<br />
2,3,7 & \left[\frac{1000}{42}\right] \:=\:23 \\ \\[-3mm]<br />
2,5,7 & \left[\frac{1000}{70}\right] \:=\:14 \\ \\[-3mm]<br />
3,5,7 & \left[\frac{1000}{105}\right] \:=\:9 \end{array}\qquad\text{Total: } 79


    . . \begin{array}{c|c}<br />
\text{Div.by} & \text{Quantity} \\ \hline \\[-3mm]<br />
2,3,5,7 & \left[\frac{1000}{210}\right] \:=\:4 \end{array} \qquad \text{total: }4


    Divisible by 2, 3, 5, or 7: . 1175 - 414 + 79 - 4 \;=\;836


    Therefore, not divisible by 2, 3, 5, or 7: .  1000 - 836 \;=\;\boxed{164}


    Anyone care to check my work . . . please?
    .
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Oct 2008
    Posts
    124
    I found a mistake in the total of divisible by 2,3 or 2,5 etc. The total is 478 not 414. Other then that, it all works out...thanks a lot.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Inclusion Exclusion
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: March 9th 2011, 10:38 AM
  2. Inclusion/Exclusion
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: January 19th 2011, 03:49 AM
  3. inclusion & exclusion help
    Posted in the Discrete Math Forum
    Replies: 11
    Last Post: May 14th 2010, 09:57 AM
  4. Inclusion/Exclusion
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: November 3rd 2009, 08:46 AM
  5. Inclusion-exclusion
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: July 28th 2009, 02:45 AM

Search Tags


/mathhelpforum @mathhelpforum