Ghosts, Ghouls and Vampire… Numbers?

October 31, 2017 9:00 am

‘Vampire’ and ‘number’ are words you might hear a lot around Spark this Halloween. Today’s spooky edition of Spark Ideas comes courtesy of our KS5 Maths lead Chris. It isn’t often that you hear that numbers can be monstrous…

 

Happy Hallowe’en everyone! I hope everyone has a great time celebrating this ghoulish holiday!

 

What? Trick or Treat, I hear you say?! Well, let me present to you a wicked trick.

 

I have been looking at a special group of numbers called vampire numbers, or simple vampires for short.

I have a number? Hell yeah!

 

What exactly is a vampire number?

 

A vampire must have an even number of digits (say n digits). They then can be divided into two integers, let’s call them x and y, both n/2 digits long. These numbers are called fangs. Quite appropriate, isn’t it?

 

The beauty of these kinds of numbers is that all digits in the original number are found in either fang exactly once.

 

Let’s dive in and look at an example!

 

The first possible vampire is 1260. It has four digits, meaning that both of its fangs will have 2 digits.

 

1260 = 21 x 60

 

We can see that all digits in 1260 can be found in its fangs, 21 and 60.

 

The next 4-digit vampire number is 1395:

 

1395 = 15 x 93

 

These are just two examples of what we call a “true” vampire.

 

One rule that we need to include is that the fangs cannot both end in zeroes. For example, if we look at 1260 and multiply both fangs by a factor of 10, we get:

 

126,000 = 210 x 600

 

Since we have both fangs have an end digit of zero, we can conclude that 126,000 is not a vampire number. So we can only have one fang ending in one zero to count.

 

Unfortunately, vampire numbers are rather difficult to find, just like in real life, thank goodness! In fact, there are only seven 4-digit examples possible:

 

1260, 1395, 1435, 1530, 1827, 2187 and 6880.

 

Can you show that these seven numbers are indeed vampires? If you’re feeling brave enough, then leave your answer in the comments section!

Are there any more complicated examples?

 

It is when we start to look at some of the 6-digit vampires when we begin to be introduced to special types. For example, you might come across a “prime” vampire number; this means both fangs are prime numbers.

 

The first prime vampire number is 117,607:

 

117,607 = 167 x 701

 

Another example of is 371,893:

 

371,893 = 383 x 971

 

This variation is only a small peep into this deep, dark crypt full of numbers, which shall, hopefully, remained nailed firmly shut until next year’s spook season arrives…

 

Until then, stay safe everyone when out and about tonight, or else your number’s up!