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2017 CNMO Grade 11 P6

Source: 2017 China Northern MO, Grade 11, Problem 6

February 24, 2020
number theory

Problem Statement

Define Sr(n)S_r(n): digit sum of nn in base rr. For example, 38=(1102)3,S3(38)=1+1+0+2=438=(1102)_3,S_3(38)=1+1+0+2=4. Prove: (a) For any r>2r>2, there exists prime pp, for any positive intenger nn, Sr(n)≡nmod  pS_{r}(n)\equiv n\mod p. (b) For any r>1r>1 and prime pp, there exists infinitely many nn, Sr(n)≡nmod  pS_{r}(n)\equiv n\mod p.
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