document.write( "Question 1029991: A \"vampire number\" is one whose digits can be regrouped into 2 smaller numbers of equal length (called \"fangs\") that multiply together to make the original number.
\n" ); document.write( "for example, 1260 is a vampire number because its digits can be regrouped to form the numbers 21 and 60 (note: both fangs cannot end in 0) and 21x60=1260\r
\n" ); document.write( "\n" ); document.write( "there are 148 vampire numbers with 6 digits. show that the following 6 digit numbers are vampire numbers \r
\n" ); document.write( "\n" ); document.write( "1) 378450
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Algebra.Com's Answer #644976 by Edwin McCravy(20065) $\"\"$ $\"About$

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A \"vampire number\" is one whose digits can be regrouped
\n" ); document.write( "into 2 smaller numbers of equal length (called \"fangs\")
\n" ); document.write( "that multiply together to make the original number.
\n" ); document.write( "For example, 1260 is a vampire number because its digits
\n" ); document.write( "can be regrouped to form the numbers 21 and 60
\n" ); document.write( "(note: both fangs cannot end in 0) and 21x60=1260\r
\n" ); document.write( "\n" ); document.write( "There are 148 vampire numbers with 6 digits. Show that
\n" ); document.write( "the following 6 digit numbers are vampire numbers \r
\n" ); document.write( "\n" ); document.write( "1) 378450
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document.write( "We must regroup the digits 3,7,8,4,5,0  into two 3-digit \r\n" );
document.write( "numbers so that their product is 378450.\r\n" );
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document.write( "Since the last digit of 378450 is 0, the last digits of\r\n" );
document.write( "the 3-digit numbers are 0 and 5\r\n" );
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document.write( "The first two digits of 378450 are 37, so we must choose \r\n" );
document.write( "the first two digits so they will multiply as close to 37 \r\n" );
document.write( "as possible without their product being more than 37.  \r\n" );
document.write( "Since 5 must be a last digit, we can't choose 5 for a first \r\n" );
document.write( "digit of either, so the only choice for the two first \r\n" );
document.write( "digits are 8 and 4.\r\n" );
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document.write( "So that narrows it down to these 4 choices:\r\n" );
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document.write( "430×875 = 376250, no\r\n" );
document.write( "470×835 = 392450, no\r\n" );
document.write( "435×870 = 378450, yes\r\n" );
document.write( "475×830 = 394250, no.\r\n" );
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document.write( "So we have shown that 378450 is a vampire number, with fangs\r\n" );
document.write( "435 and 870\r\n" );
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document.write( "2) 567648\r\n" );
document.write( "We must regroup the digits\r\n" );
document.write( "5,6,7,6,4,8  into two 3-digit numbers so that\r\n" );
document.write( "their product is  567648.\r\n" );
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document.write( "Since the first two digits of 567648 are 56, we must choose the \r\n" );
document.write( "two first digits so they will multiply as close to 56 as possible \r\n" );
document.write( "without the product being more than 56.  One of the two first\r\n" );
document.write( "digits must be 8, because if not, then 7×6 would only be 42, which \r\n" );
document.write( "is not near 56 at all. \r\n" );
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document.write( "The last digit of 567648 is 8.  Since 8 must be a first digit, we\r\n" );
document.write( "can't choose 8 for a last digit of either, so the only choice for\r\n" );
document.write( "the two last digits, so as to get last digit 8 are 4 and 7. \r\n" );
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document.write( "So we must choose the next largest digit, 6, for the other first\r\n" );
document.write( "digit.\r\n" );
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document.write( "So that narrows it down to these 4 choices:\r\n" );
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document.write( "857×664 = 569048, no\r\n" );
document.write( "867×654 = 567018, no\r\n" );
document.write( "854×667 = 569618, no\r\n" );
document.write( "864×657 = 567648, yes.\r\n" );
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document.write( "So we have shown that 567648 is a vampire number, with fangs\r\n" );
document.write( "864 and 657.\r\n" );
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document.write( "Edwin

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