Automodular

From an idea by Eric Angelini on fr.sci.maths, see his web site

These are defined as numbers N which, when split into two slices A and B are such as the remainder of dividing N by B is A.
For instance 23 = 7×3 + 2
We mainly consider cases when the remainder is the leftmost slice ("automodular to left"), and the divisor is the rightmost slice.
But we may as well consider the opposite, for instance 21 is automodular "to right" :
21 = 10×2 + 1
We won't consider empty slices, then excluding from automodularity the one digit numbers, or with A=B.
And also a number (namely B) is not allowed to begin with 0.

Smallest automodular number (to the left) with 3 different digits ?
Solution

We shall name panmodular numbers, numbers which are automodular for each possible splittings.
N = AB ≡ min(A,B) modulo max(A,B) ∀ (A,B)
Then 2 digits automodular numbers to the left or to the right are trivially panmodular (as there is only one way to split).
The smallest non trivial panmodular number is then 111, it is both automodular to the left and to the right.
111 = 10×11 + 1 (to the right)
111 = 10×11 + 1 (to the left)
We have then all possibilities of splitting 111 in two (non empty) slices.

Smallest non trivial panmodular number with all different digits ?
Solution

Searching for panmodular numbers ending in different digits is still open...
Details

 

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