document.write( "Question 1025865: How many numbers from 1-1000 are there that are not divisible by any of the digits 2,3,5. \n" ); document.write( "

Algebra.Com's Answer #641178 by ikleyn(52788) $\"\"$ $\"About$

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\n" ); document.write( "How many numbers from 1-1000 are there that are not divisible by any of the $\"highlight%28cross%28digits%29%29\"$ numbers 2,3,5.
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document.write( "500 are divisible by 2            (set A).\r\n" );
document.write( "333 are divisible by 3            (set B).\r\n" );
document.write( "200 are divisible by 5            (set C).\r\n" );
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document.write( "166 are divisible by  6 = 2*3     (set AB, intersection of A and B).\r\n" );
document.write( "100 are divisible by 10 = 2*5     (set AC, intersection of A and C).\r\n" );
document.write( " 66 are divisible by 15 = 3*5     (set BC, intersection of B and C).\r\n" );
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document.write( " 33 are divisible by 30 = 2*3*5   (set ABC, intersection of A, B, and C).\r\n" );
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document.write( "Now, the number of integers from 1 to 1000 that are not divisible by any of the numbers 2,3,5, is\r\n" );
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document.write( "  1000 - |A| - |B| - |C| + |AB| + |AC| + |BC| - |ABC|.     (1) \r\n" );
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document.write( "It is equal to\r\n" );
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document.write( "  1000 - 500 - 333 - 200 + 166 + 100 + 66 - 33 = 266.\r\n" );
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document.write( "And all that you (we, I) need is to prove the formula (1).\r\n" );
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document.write( "I will do it now.\r\n" );
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document.write( "Since we want to count those integer from 1 to 1000 that are not divisible by any of the numbers 2,3,5, \r\n" );
document.write( "we exclude/distract the quantities of |A|, |B|, and |C| from 1000.  (Step 1).\r\n" );
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document.write( "But doing this, we exclude/distract two times those integers that are in the intersections AB, AC and BC. \r\n" );
document.write( "Therefore, we add these quantities in (1) for compensation.   (Step 2). \r\n" );
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document.write( "In the step 1, we excluded/distracted 3 times the numbers of the set ABC.\r\n" );
document.write( "In the step 2, we restored/added      3 times the numbers of the set ABC.\r\n" );
document.write( "But we still need to exlude/distract the set ABC.\r\n" );
document.write( "And we do it in the last term of the formula (1).\r\n" );
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document.write( "The formula (1) is proved, and the problem is solved.\r\n" );
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document.write( "Congratulations !!!\r\n" );
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