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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.
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
there are 148 vampire numbers with 6 digits. show that the following 6 digit numbers are vampire numbers
1) 378450
2) 567648
Answer by Edwin McCravy(20064)   (Show Source):
You can put this solution on YOUR website! 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.
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
There are 148 vampire numbers with 6 digits. Show that
the following 6 digit numbers are vampire numbers
1) 378450
We must regroup the digits 3,7,8,4,5,0 into two 3-digit
numbers so that their product is 378450.
Since the last digit of 378450 is 0, the last digits of
the 3-digit numbers are 0 and 5
The first two digits of 378450 are 37, so we must choose
the first two digits so they will multiply as close to 37
as possible without their product being more than 37.
Since 5 must be a last digit, we can't choose 5 for a first
digit of either, so the only choice for the two first
digits are 8 and 4.
So that narrows it down to these 4 choices:
430×875 = 376250, no
470×835 = 392450, no
435×870 = 378450, yes
475×830 = 394250, no.
So we have shown that 378450 is a vampire number, with fangs
435 and 870
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2) 567648
We must regroup the digits
5,6,7,6,4,8 into two 3-digit numbers so that
their product is 567648.
Since the first two digits of 567648 are 56, we must choose the
two first digits so they will multiply as close to 56 as possible
without the product being more than 56. One of the two first
digits must be 8, because if not, then 7×6 would only be 42, which
is not near 56 at all.
The last digit of 567648 is 8. Since 8 must be a first digit, we
can't choose 8 for a last digit of either, so the only choice for
the two last digits, so as to get last digit 8 are 4 and 7.
So we must choose the next largest digit, 6, for the other first
digit.
So that narrows it down to these 4 choices:
857×664 = 569048, no
867×654 = 567018, no
854×667 = 569618, no
864×657 = 567648, yes.
So we have shown that 567648 is a vampire number, with fangs
864 and 657.
Edwin
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