MathDB

Ruby has a non-negative integer

n

. In each second, Ruby replaces the number she has with the product of all its digits. Prove that Ruby will eventually have a single-digit number or

0

. (e.g.

86\rightarrow 8\times 6=48 \rightarrow 4 \times 8 =32 \rightarrow 3 \times 2=6

)
Proposed by Wong Jer Ren

2