Problems & Puzzles: Puzzles

Puzzle 204.  An old empirical observation

Some of you already know that I have a file/archive of 'puzzles for the future', that contains all the puzzles suggested by the puzzlers as well as my own puzzles collected/constructed over all these years that I have been keeping this site about prime puzzles.

A very old empirical personal observation in this file simply says:

If p is a prime number, then p divides A(0)p-1B, where A is the first digit in p and B are the other digits in the same p? (I have verified this up to the prime 2593)

As a matter of fact now I have forgotten where this observation came from and if someday I solved it or not. It doesn't matter. Maybe it's better this oblivion of mine...


1. Can you explain this empirical observation?
2. Is this observation part or consequence of a more general (and less bizarre) divisibility property?

(I'm sorry if this puzzle becomes extremely easy. You can say that I'm facing the Christmas days a kind of tired... and by the way Merry Christmas to all of you, my dear friends!)

As a matter of fact this puzzle was an easy puzzle for some of you: Jon Wharf, Daniel Gronau, Jens Kruse Andersen Alireza Bakhtiari, Joseph L. Pe, Johann Wiesenbauer and Ken Wilke. All of them wrote - more or less - the same, like this:

Your empirical observation follows immediately from Fermat's Little Theorem. Suppose that p is a prime which contains k digits where k >= 1. Then p = 10^(k-1)*A + B.

Then A(0)(p-1)B, ( read as A followed by p-1 zeros followed by B), = A*(10^{(p - 1)+(k - 1)) + B = A*{10^(k-1)}*{10^(p-1)-1} + A*(10^(k-1)) + B = A*{10^(k-1))*{10^(p-1)-1)} + p which is divisible by p since Fermat's Little Theorem guarantees that p divides the quantity -1 + 10^(p - 1) since gcd(10,p) = 1. Q. E. D.








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