# 2021-01 Single-digit number

Prove that for any given positive integer $$n$$, there exists a sequence of the following operations that transforms $$n$$ to a single-digit number (in decimal representation).

1) multiply a given positive integer by any positive integer.

2) remove all zeros in the decimal representation of a given positive integer.

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# 2012-18 Diagonal

Let $$r_1,r_2,r_3,\ldots$$ be a sequence of all rational numbers in $$(0,1)$$ except finitely many numbers. Let $$r_j=0.a_{j,1}a_{j,2}a_{j,3}\cdots$$ be a decimal representation of $$r_j$$. (For instance, if $$r_1=\frac{1}{3}=0.333333\cdots$$, then $$a_{1,k}=3$$ for any $$k$$.)

Prove that the number $$0.a_{1,1}a_{2,2}a_{3,3}a_{4,4}\cdots$$ given by the main diagonal cannot be a rational number.

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