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Part of 2024 Centroamerican and Caribbean Math Olympiad

Problems(1)

Alero Numbers in the Centro

Source: Centroamerican and Caribbean Math Olympiad 2024 P1

10/17/2024
Let nn be a positive integer with kk digits. A number mm is called an aleroalero of nn if there exist distinct digits a1a_1, a2a_2, ⋯\dotsb, aka_k, all different from each other and from zero, such that mm is obtained by adding the digit aia_i to the ii-th digit of nn, and no sum exceeds 9. For example, if nn == 20242024 and we choose a1a_1 == 22, a2a_2 == 11, a3a_3 == 55, a4a_4 == 33, then mm == 41774177 is an alero of nn, but if we choose the digits a1a_1 == 22, a2a_2 == 11, a3a_3 == 55, a4a_4 == 66, then we don't obtain an alero of nn, because 44 ++ 66 exceeds 99. Find the smallest nn which is a multiple of 20242024 that has an alero which is also a multiple of 20242024.
OMCCnumber theory