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4

Part of 2018 OMMock - Mexico National Olympiad Mock Exam

Problems(1)

Divisibilities on sum of digits

Source: Mexico National Olympiad Mock Exam 2018 Problem 4

11/6/2018
For each positive integer nn let s(n)s(n) denote the sum of the decimal digits of nn. Find all pairs of positive integers (a,b)(a, b) with a>ba > b which simultaneously satisfy the following two conditions
a∣b+s(a)a \mid b + s(a) b∣a+s(b)b \mid a + s(b)
Proposed by Victor Domínguez
number theorysum of digitsDivisibility