518 Repunits
18.1 k-right-transposable integers ...........
Letkbe a given positive integer. A positive integerXisk-transposable
if in moving the leftmost digit to the rightmost, the number is multiplied
byk.
Note thatXis a repdigit if and only ifk=1. We shall assumek> 1.
Suppose the numberXhasndigits, with leftmost digita.Wehave
10(X−a· 10 n−^1 )+a=kX.
From this,(10−k)X=a(10n−1) = 9a·Rn.
Ifk=3, 10 −kcan only have prime divisors 2, 3, 5. The equation
will reduce toX=a repdigit, which is clearly impossible.
Fork=3,wehave 7 X=a(10n−1).Ifa=7, then againXis a
repdigit. Therefore, we must have 7 dividing 10 n− 1. This is possible
only ifnis a multiple of 6. ThereforeX=a·^10
6 m− 1
7 and has first digit
a.
Now,
106 m− 1
7
= (142857)m.
It is easy to see thatacan only be 1 or 2.
Therefore, the onlyk-transposable numbers are(142857)mand(285714)m
withk=3.
