## Just another number: 526315789473684210! A multiplication trick!!

Yesterday, Gaurav (@giitjee) posted on his blog about a very interesting number happened to him, i.e. 381654729.  This post is as a followup of the little conversation we had in the comment section there.

There is a simple trick with number 526315789473684210.  Once you learn and remember this number, you can multiply it by any number less than 19 just within seconds.  It would take more time to write down the number than the actual calculation.  For example:

526315789473684210 x 17 = 8947368421052631570

526315789473684210 x 13 = 6842105263157894730

The trick is simple:

• Multiply by 5 to the number by which we are multiplying.  In our first example, 17 x 5 = 85.
• Look for the the number just a little greater than 85 in the original example; that is 89.
• Write the number starting from there ahead: 89473684210.
• Add the previous part of the number subsequent to it: “89473684210”5263157″
• Add a zero at the end: 8947368421052631570

Try it yourself and leave me a feedback!!

Any proof?
I don’t know why does it happen.  I know the number is deduced as a reciprocal of 19; 1/19=0.05263157894736842…. It must have something to do with the trick.  It also happens similarly with the reciprocal of 13, just we need to multiply by 6 and look for a number little lesser in that case.

Leave me a note if you know the proof or it is already somewhere on the net.  Thanks 🙂

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### 4 Responses to “Just another number: 526315789473684210! A multiplication trick!!”

1. Gaurav Tiwari Says:

I’m glad that I could inspire you! These type of numbers are called Cyclic Numbers. A cyclic number is an $n$-digit integer that, when multiplied by $\mathrm{1, 2, 3, ...,n}$ produces the same digits in a different order. Cyclic numbers are generated by the full reptend primes, i.e., 7, 17, 19, 23, 29, 47, 59, 61, 89, 97, ….|
Nicely written!

2. Ganesh Dhamodkar Says:

Thanks Gaurav! I need to read more about reptend prime numbers and cyclic numbers!

3. Gaurav Tiwari Says:

Wikipedia has a better article on this topic: http://en.wikipedia.org/wiki/Cyclic_Number/
To be honest, I was not aware about these terms. It’s you who led me to research about them.

4. Andre Says:

This number has a particularity. It is made of 18 digits, 16 of which is a duplication of all numbers from1 to 8 + 9 Andre 0 once each. Each number is once followed once by a number greater than itself and once by a number smaller than it self. Try to multiply this number by a multiple of 19!!!