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Congruency in sum of digits base q

Source: China Team Selection Test 2016 Test 3 Day 2 Q4

March 26, 2016
number theory

Problem Statement

Let a,b,b′,c,m,qa,b,b',c,m,q be positive integers, where m>1,q>1,∣b−b′∣≥am>1,q>1,|b-b'|\ge a. It is given that there exist a positive integer MM such that Sq(an+b)≡Sq(an+b′)+c(modm)S_q(an+b)\equiv S_q(an+b')+c\pmod{m}
holds for all integers n≥Mn\ge M. Prove that the above equation is true for all positive integers nn. (Here Sq(x)S_q(x) is the sum of digits of xx taken in base qq).
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