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Number Theory

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Prime Related Number: Rationality Determination

NUMTH-ADWRHH

We construct a number $\alpha$ in decimal by the following rules:

Rule 1: The number $\alpha$ starts with the first digit $0$, i. e. its decimal expansion is $0. ***\cdots $.

Rule 2: The $n$-th digit of $\alpha$ after the decimal point is $0$ if $n$ is not a prime, and it is $1$ if $n$ is a prime.

Which of the following statement is true? Select ONE answer.

A

$\alpha$ is rational.

B

$\alpha$ is irrational.

C

It is impossible to determine that $\alpha$ is rational or irrational.