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The magic number 25641

By Christian Lawson-Perfect. Posted October 27, 2016 in cp's mathem-o-blog

Reader of the site Bhaskar Hari Phadke has written in to tell us this fun fact about the number $25641$ . It’s easier to show than to describe, so here goes:

$\begin{aligned} 25641 \times 1 \times 4 & = 102564 \\ 25641 \times 2 \times 4 & = 205128 \\ 25641 \times 3 \times 4 & = 307692 \\ 25641 \times 4 \times 4 & = 410256 \\ 25641 \times 5 \times 4 & = 512820 \\ 25641 \times 6 \times 4 & = 615384 \\ 25641 \times 7 \times 4 & = 717948 \\ 25641 \times 8 \times 4 & = 820512 \\ 25641 \times 9 \times 4 & = 923076 \end{aligned}$

A good one to challenge a young person with.

I did a little bit of Sloanewhacking and found a couple of sequences containing $25641$ which almost, but don’t quite, describe this property. So, semi-spoiler warning: you might enjoy A256005 and A218857. I’d like to come up with the ‘magic number’ which looks the least like it’ll have this property – any ideas?

Thanks, Bhaskar!

4 Responses to “The magic number 25641”

Billy October 28th, 2016

I must be missing something, all I see is that the leading digit of the product is the same as the digit being multiplied.

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- Billy October 28th, 2016
  
  Disregard that, I see it now
  
  Reply
John McGough October 28th, 2016

That property applies to any 5 figure number where the first 2 figures are 25

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Marvolo Avendord October 28th, 2016

For some integer n, any number between $10^{n} / 40$ and $10^{n} / 36$ will have this property, so it’s all a matter of which one of those you think will look the least special. One that is interesting is 27601:

$27601 \times 7 \times 4 = 772828$ for example, which is nice because $4 \times 7 = 28$ . Similarly, $27601 \times 3 \times 4 = 331212$ .

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The magic number 25641

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