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Product of digits

Source: JOM 2023 P2

February 20, 2023
algebra

Problem Statement

Ruby has a non-negative integer nn. 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 00. (e.g. 86→8×6=48→4×8=32→3×2=686\rightarrow 8\times 6=48 \rightarrow 4 \times 8 =32 \rightarrow 3 \times 2=6)
Proposed by Wong Jer Ren
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