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In this work initial numbers and repunit numbers have been studied. Allnumbers have been considered in a decimal notation. The problem of simplicityof initial numbers has been studied. Interesting properties of numbers repunitare proved:  $gcd(R_a, R_b) = R_{gcd(a,b)}$;  $R_{ab}/(R_aR_b)$ is an integer only if $gcd(a,b) = 1$, where $a\geq1$,$b\geq1$ are integers. Dividers of numbers repunit, are researched by a degreeof prime number.

The concrete theory of numbers: initial numbers and wonderful properties of numbers repunit