document.write( "Question 630218: How many numberrrs less than 10000 have the product of their digits equal to 84? \n" ); document.write( "
Algebra.Com's Answer #396779 by AnlytcPhil(1806) $\"\"$ $\"About$
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document.write( "I will assume by \"number\" you mean positive integer.\r\n" );
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document.write( "Break 84 into its prime factors.\r\n" );
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document.write( "84 = 2*2*3*7\r\n" );
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document.write( "I. First we will get all possible 4 digit integers with product of digits 84:\r\n" );
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document.write( "A. Any 4 digit permutation of the digits 2,2,3,7 will have product 84\r\n" );
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document.write( "   There are 2 indistinguishable digits, so the number of distinguishable\r\n" );
document.write( "   permutations of those is  =  = 12\r\n" );
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document.write( "B. We can multiply factors 2*2 and get the digit 4 and introduce 1 to \r\n" );
document.write( "   get 4-digit integers with digits 4,3,7,1\r\n" );
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document.write( "   There are 4! or 24 permutations of these\r\n" );
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document.write( "C. We can multiply factors 2*3 and get the digit 6 and introduce 1 to \r\n" );
document.write( "   get 4-digit integers with digits 6,2,7,1\r\n" );
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document.write( "   There are 4! or 24 permutations of these\r\n" );
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document.write( "So there are 12+24+24 = 60 four-digit integers with product of digits 84.\r\n" );
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document.write( "II.  Next we will get all possible 3-digit integers with product of digits 84.\r\n" );
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document.write( "A. We can multiply factors 2*2 and get the digit 4.  So any three-digit \r\n" );
document.write( "   integer with digits 4,3,7 will have product of digits 84\r\n" );
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document.write( "   There are 3! or 6 permutations of these\r\n" );
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document.write( "B. We can multiply factors 2*3 and get the digit 6. \r\n" );
document.write( "   So any three-digit integers with digits 6,2,7\r\n" );
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document.write( "   There are 3! or 6 permutations of these\r\n" );
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document.write( "So there are 6+6 = 12 three-digit integers with product of digits 84\r\n" );
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document.write( "III.  There are no 2-digit integers with product of digits 84, since the\r\n" );
document.write( "      largest product of digits is for 99, which has product of digits of\r\n" );
document.write( "      only 81. \r\n" );
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document.write( "Answer:  60 four-digit integers and 12 three-digit integers.\r\n" );
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document.write( "         Total 72.\r\n" );
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document.write( "[Note: If you actually mean numbers, and not just integers, you would have\r\n" );
document.write( " to count each four-digit number like, say, 2347 five times, as \r\n" );
document.write( " .2345, 2.347, 23.47, 234.7, and 2347.  So there would be 60×5 or 300 four\r\n" );
document.write( " digit numbers.  Also each 3-digit number like , say, 437 four times, as \r\n" );
document.write( " .437, 4.37, 43.7, and 437.  So there would be 12×4 or 48 three-digit numbers.\r\n" );
document.write( " And in that case the total number would be 348.  However I believe you\r\n" );
document.write( " meant just integers.]      \r\n" );
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document.write( "Edwin
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