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Problem of the Week 1135

Vampire Numbers

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A "vampire number" is an integer with 2n digits that is the product of two n-digit numbers — the fangs — whose digits, when combined, form a permutation of the original digits (thus, multiplicity counts). The smallest vampire numbers are 1260 (= 21 × 60) and 1395 (= 15 × 93).

Technical point: The fangs cannot each end in a 0. So 126000 is not a vampire number.

You might have missed it, but October 5, 2010, was a vampire day because 10 05 2010 is an 8-digit vampire number. It equals 2010 × 5001.

When is the next vampire day?

What is the first double vampire number? In other words, what is the first number having two different factorizations into fangs, X = A B = C D, where A and B permute the digits of X, as do C and D?

Source: Ed Pegg's blog.

© Copyright 2010 Stan Wagon. Reproduced with permission.

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12 October 2010