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China Northern Mathematical Olympiad, Problem 7

Source: China Northern Mathematical Olympiad

July 29, 2017
number theorysum of digits

Problem Statement

Let S(n)S(n) denote the sum of the digits of the base-10 representation of an natural number nn. For example. S(2017)=2+0+1+7=10S(2017) = 2+0+1+7 = 10. Prove that for all primes pp, there exists infinitely many nn which satisfy S(n)≡nmod  pS(n) \equiv n \mod p.
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