A 2-digit Difference with Equal Digits

Given 12 distinct 2-digit integers. Prove there are some two whose difference - a 2-digit number - has equal digits.

Solution


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Copyright © 1996-2018 Alexander Bogomolny

Given 12 distinct 2-digit integers. Prove there are some two whose difference - a 2-digit number - has equal digits.

There are 11 residues of division by 11. There are 12 numbers. By the pigenhole principle, some two have the same remainder of division by 11. There difference is divisible by 11 but could not be 0 since all 12 given numbers are distinct. Other than that, the difference is a 2-digit number divisible by 11 and thus having equal digits.


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