Unique Primes
Number Theory
Level
3
There exists a unique prime with its reciprocal having period of 196. Given that p > 10^80, compute the remainder when p is divided by 10^9.
There does not exist a unique prime with period of 294
1009
9009
9
1
900900101
9001001
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Note: These kind of primes are actually called 'Unique Primes' whose reciprocal has a period of k which no other prime has. This problem was a kind in OMO Spring'19, but instead of 196, it was 294.
Solution: Every Unique Prime can be written (Phi n(10))/(gcd(Phi n(10),n))=p^alpha, where Phi n(10) is a cyclotomic polynomial. Now, I would like you (reader) to find Phi 196(10) and then find the gcd in the denominator and then solve for congruence of Phi 196(10) with 10^9. Then continue with the equation of gcd in the denominator and Phi 196(10). I hope this helps.