What Positive Number Doubles When The Last Digit Moves To The First Digit? Riddle For "Geniuses"
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users cite it as the source of this digit-shifting puzzle and as a starting point for their own algebraic, modular arithmetic and multiplicative-order write-ups.
Recommended by 1 people across 1 different months, 2017 to 2020.
What it is
A MindYourDecisions video by Presh Talwalkar on the 'parasitic number' puzzle: find the smallest positive whole number that doubles when its last digit is moved to the front. It walks through the algebra and the long division pattern that produces the 18-digit answer, and mentions Freeman Dyson's reported instant answer about the digit count.
What people said
Recommends Smer5 · 2020-08 · Mathematics Stack Exchange · on “A number Doubles When Last Digit Moves To The First Digit: Fermats Little theorem usage.”
My addition to the solution from here [this video] goes whith usage of Fermats little theorem, but i want ask, if it is used properly there, because i have some doubts. But if it is correct, the solution is really short and elegant. So the task is to find the least positive number, what doubles, if you move the least digit - the right digit to the first - on very left position. So let a be the first digit in [1..9] b the rest m - amount of digits after the first one $$ \begin{cases} x = 10 b + a \\ 2 x = a *10^m + b\end{cases} \\ 20b + 2a = a *10^m + b \\ b = \frac{2a*(5 * 10^{m-1}-1) }{19 }$$ Since b is natural number, and a is a digit [1..9], the rest in brackets must be divisible to 19, or: $$
Cites user377597 · 2017-07 · Mathematics Stack Exchange · on “How is moving the last digit of a number to the front and multiplying related to multiplicative orders?”
There seems to be a relationship between multiplicative orders modulo $n$ [LINK] and a puzzle Presh Talwalkar gave a few days ago at [this video] I'm hoping someone can give a more rigorous explanation of the pattern I see. $\bullet\textbf{ The Puzzle }\bullet$ He states the puzzle as: "What Positive Number Doubles When Its Last Digit Becomes Its First Digit?" For example, the smallest solution - assuming base-10 representation - is $105263157894736842$. Which means that $$(2)\cdot105263157894736842 \quad\ \ \ $$ $$||$$ $$210526315789473684$$ Naturally, one can proceede to find solutions in other bases. The base-5 solution is $$(2)\cdot102342_5 \quad\ \ \ $$ $$||$$ $$210234_5$$ $\bullet\textbf{ M
Cites guy600 · 2018-07 · Mathematics Stack Exchange · on “Finding numbers that double when you switch the first and last digit”
So, I was watching this video [this video] by MindYourDecisions about finding the smallest number that doubles when you move the last digit to become the first digit. Actually, I just saw the title of the video, and I found it interesting, so I began to work on a solution. But, in fact, I had misinterpreted the question posed by the video. I was looking for a number that doubles when you switch the first and last digits of a number. I started off my search by writing equations like: $$ 2(10a + b) = 10b + a $$ This simplifies to: $$ 19a = 8b$$ Since $a$ and $b$ should be integers between 0 and 9 inclusive, I figured out that there are no solutions amongst two-digit numbers. Next, I applied this te