document.write( "Question 1137732: How many numbers between 100 and 1000 are
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Algebra.Com's Answer #755603 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( " The formulation of the problem in the post leaves the room for questions\r
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document.write( "                is 100 included ?    Is 1000 included ?\r\n" );
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\n" ); document.write( "\n" ); document.write( " In this sense, this formulation is unprofessional. A professional formulation of a Math problem does not leave \r
\n" ); document.write( "\n" ); document.write( " the room for such questions. Therefore, I will reformulate the problem in this way:\r
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document.write( "            How many three-digit numbers are\r\n" );
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document.write( "              - Not divisible by 2 ?\r\n" );
document.write( "              - Not divisible by 3 ?\r\n" );
document.write( "              - Not divisible by either 2 or 3 ?\r\n" );
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\n" ); document.write( "\n" ); document.write( " Thre-digit numbers are the numbers from 100 to 999 inclusively, so there is no uncertainty with this formulation.\r
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\n" ); document.write( "\n" ); document.write( "(a) How many three-digit numbers are not divisible by 2 ?\r
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document.write( "     Every second integer number in the interval [100,999] is divisible by 2.\r\n" );
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document.write( "     The number of such pairs is   =  = 450.\r\n" );
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document.write( "     So, 450 of the 900 numbers are divisible by 2, and the rest, 900-450 = 450 ARE NOT divisible by 2.    ANSWER\r\n" );
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\n" ); document.write( "\n" ); document.write( "(b) How many three-digit numbers are not divisible by 3 ?\r
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document.write( "     Every third integer number in this interval is divisible by 3.\r\n" );
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document.write( "     More precisely, every third, starting from 102.\r\n" );
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document.write( "     The number of such triples is   = 299.\r\n" );
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document.write( "     To it, I must add 1 to account for the number 999, which goes individually, without companions.\r\n" );
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document.write( "     So, 300 = 299+1 of the 900 numbers are divisible by 3, and the rest, 900-300 = 600 ARE NOT divisible by 3.    ANSWER\r\n" );
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\n" ); document.write( "\n" ); document.write( "(c) How many three-digit numbers are not divisible by either 2 or 3 ?\r
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document.write( "     As a first approach, we can subtract 450 and 300 from 900 - those integer numbers that are divisible by 2 and by 3.\r\n" );
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document.write( "     900 - 450 - 300 = 150.\r\n" );
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document.write( "     But doing in this way, we subtract multiples of 6 twice (!).\r\n" );
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document.write( "     Therefore, we must return back the number of multiples of 6 among 3-digit numbers.\r\n" );
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document.write( "      Again, we need to calculate the number of segments of the length 6 from 100 to 999 inclusively.\r\n" );
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document.write( "       = 150.  \r\n" );
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document.write( "      Hence, the number of multiples to 6 between 100 and 999 is 150.\r\n" );
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document.write( "     Therefore, our final answer to question (c) is  150 + 150 = 300.      ANSWER\r\n" );
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