Question 1111064
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<pre>
I will consider integer numbers from 201 to 500 <U>INCLUSIVELY</U>.


The number of such integer numbers is 500-200 = 300.   (<U>It is VERY good number,</U>
                                                        <U>since it is divisible by any of 4, 6, 10, 12, 20, 30, 50, 60 . . . </U>)


Of these 300 numbers, 

    there are 300/4 = 75 numbers divisible by 4.    Let me denote this set of numbers as Z4.

              300/6 = 50 numbers divisible by 6.    Let me denote this set of numbers as Z6.

              300/10 = 30 numbers divisible by 10.  Let me denote this set of numbers as Z10.

              300/12 = 25 numbers divisible by 12.  Let me denote this set of numbers as Z12.

    Now I need to find the number of elements in the union set  Z4 U Z6 U Z10 U Z12.


    I can easily simplify my task by noticing that  Z12  is just included into  Z4 U Z6  (as their intersection !).


    So, all I need is to find the number of elements in the union set  Z4 U Z6 U Z10.


    Now I will use THIS STATEMENT:

    If A, B and C are <U>finite</U> sub-sets of the universal set U, then


        |A U B U C| = |A| + |B| + |C| - |A n B| - |A n C| - |B n C| + |A n B n C|.  (*)


    Here |X| denotes the number of elements in a finite subset X.   (And the symbol "n" denotes the intersection of sub-sets).


    I will not prove this statement here (although it is absolutely elementary).
    I will refer you to my lessons  in this site
         <A HREF=http://www.algebra.com/algebra/homework/word/misc/Counting-elements-in-sub-sets-of-a-given-finite-set.lesson>Counting elements in sub-sets of a given finite set</A>
         <A HREF=https://www.algebra.com/algebra/homework/word/misc/Advanced-probs-counting-elements-in-sub-sets-of-a-given-finite-set.lesson>Advanced problems on counting elements in sub-sets of a given finite set</A>
    instead.


    Now, from the formula (*)


        |Z4 U Z6 U Z10| = |Z4| + |Z6| + |Z10| - |Z4 n Z6| - |Z4 n Z10| - |Z6 n Z10| + |Z4 n Z6 n Z10|.    (**)


    Here |Z4|, |Z6| and |Z10| are just known to you numbers 75, 50 and 30.

    |Z4 n Z6| is the number of those integers from 201 to 500 that are divisible by 12; this number is 300/12 = 25.

    |Z4 n Z10| is the number of those integers from 201 to 500 that are divisible by 20; this number is 300/20 = 15.

    |Z6 n Z10| is the number of those integers from 201 to 500 that are divisible by 30; this number is 300/30 = 10.

    Finally, |Z4 n Z6 n Z10| is the number of those integers from 201 to 500 that are divisible by 60; this number is 300/60 = 5.


    Therefore, the formula  (**) becomes


        |Z4 U Z6 U Z10| = 75 + 50 + 30 - 25 - 15 - 10 + 5 = 110.


    Thus the number of those integer between 201 and 500 that are divisible by  4, 6, 10  and  12  is   110.
</pre>


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