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[learning notes] number theory thinking problem

~~In fact, it's the question that broke~~

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Ingenious number theory problem
Let the length of the loop section be $l$ , The length of the non circulating part is $c$
because $b$ It's the result of the score reduction
So there is $b|10^{l+c}-10^c$
Write in congruence form $10^{l+c}\equiv 10^c\pmod b$
If $(10, b) = 1$
that $10^l\equiv 1\pmod b$
that $l=ord_b(10)$
And there is no non cyclic part of the decimal
If $(10,b)\ne 1$
set up $b=2^{k_2}5^{k_5}b'$
that $10^l\equiv 1\pmod {\frac{b}{\gcd(b,10^c)}}$
Notice that two numbers that are not coprime have no order
prove ： hypothesis $a^b\equiv 1\pmod m$ , set up $d=\gcd(a,m)$
that $a^b\equiv 1\pmod d$
because $a ∣ d$ , So left congruence $0$ , Obviously not
therefore $10^l\equiv 1\pmod {b'}$
that $l=ord_{b'}(10)$
Obviously, the non circulating part $c=\max(k_2,k_5)$
Of course , If $b^{'} = 1$ Then the decimal has no cyclic section （ In other words, irrational numbers

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