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Theory of numbers: Is each cousin contained in the set $ , {n in mathbb {N}: n | 10 ^ k-1 } cup {2,5 } , $ for $ k = 1, 2,3, … $?

It is each cousin contained in the set
$ , {n in mathbb {N}: n | 10 ^ k-1 } cup {2,5 } , $ for $ k = 1,2,3, … $?

Leave $ p $ be a prime number Then $ frac {1} {p} $ has a maximum decimal period of length $ p-1 $. Denote $ frac {1} {p} $ by $$ frac {1} {p} = 0. overline {d_1d_2 … d_r} $$ where $ 1 leq r lt p $ Y $ d_i in {0,1,2, …, 9 } $.

Leave $ d = 0.d_1d_2 … d_r cdot10 ^ r ; $ (For example, yes $ p = 7 $, so $ frac {1} {7} = 0. overline {142857} $, $ r = 6 $Y $ d = 142857 $).

Let's consider $ frac {1} {10 ^ m-1} $ for some $ m en mathbb {N} $:
$$
frac {1} {10 ^ m-1} = frac {1} { underbrace {99 … 9} _ { text {$ m $ times}}} = 0. overline { underbrace {00 … 0} _ {m-1 \ text {times}} 1} = sum_ {i = 1} ^ infty frac {1} {10 ^ {i cdot m}}
$$

So
begin {align}
frac {1} {p} & = 0. overline {d_1d_2 … d_r} = d cdot left ( frac {1} {10 ^ r} right) + d cdot left ( frac) {1} {10 ^ {2r}} right) + d cdot left ( frac {1} {10 ^ {3r}} right) + cdots \ & = d cdot left ( frac) {1} {10 ^ r} + frac {1} {10 ^ {2r}} + frac {1} {10 ^ {3r}} + cdots right) \ & = d cdot sum_ { i = 1} ^ infty frac {1} {10 ^ {i cdot r}} \ & = frac {d} {10 ^ r-1}
end {align}

Reorganizing the terms, we have $ , 10 ^ r-1 = p cdot d , Longrightarrow , p | 10 ^ r-1 $.

Does this prove that the set of prime numbers is contained in the set? $ {n in mathbb {N}: n | 10 ^ k-1 } cup {2,5 } $?

Clearly, $ 2 $ Y $ 5 $ missing in $ {n in mathbb {N}: n | 10 ^ k-1 }. Are there other cousins ​​not included in this set?