Lonely fairy in a box "infinitizing" your digits. But, how?

Hei kaikille,

This time I was thinking about doing some fun with infinitely repeating (looping) decimal digits. It's quite straightforward, and at the same time an interesting "trick" you can present to any friend just using any crappy simple calculator. :)

Of course, it is advisable to solve the problem first - in order to get assured you know what to do. (The task is not a big issue, I also wonder what rating it will get from the reviewer crew, it will surely fit in one of the basic math categories.)

So, down to details.

We're about to build a Number-Infinitizer! It is actually also a Number-and-Sum Infinitizer , anyway, let's see first, what kind of numbers do we need to test such a "device"...

I define some symbols as shorthands for any and non-zero decimal digits:

$D_0 = [0...9]$ .

$D_1 = [1...9]$ .

First, take a 2-digit number $ab : a \in D_1, b \in D_0$ in a fashion that:

• The number is not less than 10. (There is no leading zero.)

• Even the sum of the two digits $a + b$ doesn't exceed $9$ - simply put, the sum is also a single digit, just like $a$ and $b$ . Let this "sum" digit be named $s \in D_1$ .

Note: In this problem, concatenated letters , e.g. the term $ab$ doesn't denote $a \times b$ , treat them rather as concatenated digits:

• e.g. : $ab = a \times 10 + b$ . The same holds for all similar situations.

$ab$ could be, let's say, 21. Or maybe 54, 25, 81, 90, etc... $a$ and $b$ may also hold the same digit (e.g. 33, 44), but then the results will look less "impressive" (however, will still work!). Some invalid numbers: 05 (less than ten; leading zero present), 37 (sum rule violation - sum (10) is not a single digit), 99 (sum rule problem - sum (18) is not a single digit).

That's for the intro part.

Now, imagine an "Infinitizer Machine", InfM , such that:

• $\mathbf{InfM(ab) = ab.asbasbasbasbasbasb...}$ (infinitely looping decimals) ,

with $ab$ being the 2-digit number of your choice and $s$ being their sum digit.

Some straight examples:

• $\mathbf{InfM(53) = 53.583583583583583583... ,}$

• $\mathbf{InfM(14) = 14.154154154154154154... ,}$

• etc....

Now, let's take a closer look at the "Machine" itself. Well, inside the casing there's nothing overly much, just a small fairy sitting there alone, with the sole magical skill of performing the following simple function mindblowingly fast, all-day and all-night:

$\mathbf{InfM(x) = x \times \frac{p}{q}} : p, q \in \mathbb{Z}^{+}.$

So, the box has been demistified, and this is the point where my turn is over - and you're in to catch the steering wheel and drive on!

The task: Assign values to $p$ and $q$ in a way that they make $InfM(x)$ , hence also the machine working exactly as it was demonstrated on the above examples, with any valid $ab$ given as input. I think you'll get through this without much hassle.

With the two values in your hands, take the fraction $\frac{p}{q}$ , simplify it the to the simplest form (if necessary), so that $p$ and $q$ are relative primes (have a GCD of 1). Then calculate $p \times q$ , type in the form and send! This will be your numeric solution. :)

Thereafter, unless you're completely surrounded by colleagues and friends of solely math post-docs, beautiful minds, rainmen and (good) Will Hunting in person, you may even have some chance to show the infinitizing trick to some friends as well... :)

Y.t.,

Ꮙesa :)