MathDB

A finite sequence of decimal digits from

\{0,1,\cdots, 9\}

is said to be common if for each sufficiently large positive integer

n

, there exists a positive integer

m

such that the expansion of

n

in base

m

ends with this sequence of digits.
For example,

0

is common because for any large

n

, the expansion of

n

in base

n

10

, whereas

00

is not common because for any squarefree

n

, the expansion of

n

in any base cannot end with

00

.
Determine all common sequences.
Proposed by Wong Jer Ren

Problem Statement