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numbers with the same digits

Source: Polish Junior Olympiad 2021 Final Round Problem 5, https://omj.edu.pl

March 21, 2021

DigitsDivisibilitynumber theory

Problem Statement

Natural numbers

a

b

are written in decimal using the same digits (i.e. every digit from 0 to 9 appears the same number of times in

a

and in

b

). Prove that if

a+b=10^{1000}

then both numbers

a

and

b

are divisible by

10