18.2k-left-transposable integers 519
18.2 k-left-transposable integers ...........
Letkbe a given positive integer. A positive integerXisk-left-transposable
if in moving the rightmost digit to the leftmost, the number is multiplied
byk. Note thatXis a repdigit if and only ifk=1. We shall assume
k> 1. Suppose the numberXhasndigits, and its rightmost digit isb.
We h ave
b· 10 n−^1 +X−b
10=kX.From this,b· 10 n+X−b=10kX, and
(10k−1)X=b(10n−1).SinceXis hasndigits,b(10n−1)>(10k−1)10n−^1 , andb>^10
n− (^1) (10k−1)
10 n− 1 =
k− 101. This shows thatb≥k.
kn
219 X=b(10n−1) 18m
329 X=b(10n−1) 28m
439 X=b(10n−1) 12m
549 X=b(10n−1) 42m
659 X=b(10n−1) 58m
769 X=b(10n−1) 22m
879 X=b(10n−1) 13m
989 X=b(10n−1) 44m
These lead to the following numbers:
X 2 =105263157894736842,
X 3 =1034482758620689655172413793,
X 4 =102564102564,
X 5 =102040816326530612244897959183673469387755,
X 6 =1016949152542372881355932203389830508474576271186440677966,
X 7 =1014492753623188405797,
X 8 =1012658227848,
X 9 =10112359550561797752808988764044943820224719.
Each of theseXkcan be replaced bykb·Xkfork=b,..., 9. Every
k-left-transposable number is of the form(Xk)mforXkgiven above and
m≥ 1.