Question 1137732
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            The formulation of the problem in the post leaves the room for questions


<pre>
                is 100 included ?    Is 1000 included ?
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;In this sense, this formulation is unprofessional. &nbsp;A professional formulation of a Math problem does not leave 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the room for such questions. &nbsp;Therefore, &nbsp;I will reformulate the problem in this way:


<pre>
            How many three-digit numbers are

              - Not divisible by 2 ?
              - Not divisible by 3 ?
              - Not divisible by either 2 or 3 ?
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Thre-digit numbers are the numbers from 100 to 999 inclusively, so there is no uncertainty with this formulation.



(a)  &nbsp;&nbsp;<U>How many three-digit numbers are not divisible by 2</U> ?


<pre>  
     Every second integer number in the interval [100,999] is divisible by 2.

     The number of such pairs is  {{{(999-99)/2}}} = {{{900/2}}} = 450.

     So, 450 of the 900 numbers are divisible by 2, and the rest, 900-450 = 450 ARE NOT divisible by 2.    <U>ANSWER</U>
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(b)  &nbsp;&nbsp;<U>How many three-digit numbers are not divisible by 3</U> ?


<pre>
     Every third integer number in this interval is divisible by 3.

     More precisely, every third, starting from 102.

     The number of such triples is  {{{(998-101)/3}}} = 299.

     To it, I must add 1 to account for the number 999, which goes individually, without companions.

     So, 300 = 299+1 of the 900 numbers are divisible by 3, and the rest, 900-300 = 600 ARE NOT divisible by 3.    <U>ANSWER</U>
</pre>


(c)  &nbsp;&nbsp;<U>How many three-digit numbers are not divisible by either 2 or 3</U> ?


<pre>
     As a first approach, we can subtract 450 and 300 from 900 - those integer numbers that are divisible by 2 and by 3.

     900 - 450 - 300 = 150.

     But doing in this way, we subtract multiples of 6 twice (!).

     Therefore, we must return back the number of multiples of 6 among 3-digit numbers.

      Again, we need to calculate the number of segments of the length 6 from 100 to 999 inclusively.

      {{{(999-99)/6}}} = 150.  

      Hence, the number of multiples to 6 between 100 and 999 is 150.

     Therefore, our final answer to question (c) is  150 + 150 = 300.      <U>ANSWER</U>
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Solved.