Tag Archives: decimal

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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