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ASU 085 All Soviet Union MO 1967 rearranging digits of a natural

Source:

June 20, 2019
number theoryDigits

Problem Statement

a) The digits of a natural number were rearranged. Prove that the sum of given and obtained numbers can't equal 999...9999...9 (19671967 of nines).
b) The digits of a natural number were rearranged. Prove that if the sum of the given and obtained numbers equals 10101010, than the given number was divisible by 1010.