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, 2006

Algebra Difficulty 5.2 AIME, harder Prove it Italy

The members of a tribe have ten fingers on their hands and nine on their feet, and therefore they count indifferently in base 10 or 19. In their mathematical culture, a positive integer is called "sacred" if in both bases it is written with the same two digits (between 1 and 9). How many sacred numbers are there?

Solution

Solution:

The answer is 4. Let $n$ be a sacred number and let $A B$ be the writing of the number in base 10. Then in base 19, by sacredness, it is written either as $A B$ or as $B A$ . The first case generates no solutions since $A$ and $B$ are different from 0, and therefore $A B$ in base 19 is greater than $A B$ in base 10 for every $A, B$ . One must then have (writing in base 10) $10A + B = 19B + A$ , that is $9A = 18B$ , from which $A = 2B$ . Since $10 > A > 0, 10 > B > 0$ , the allowed values for $B$ are then $1, 2, 3, 4$ , which give rise to the numbers $21, 42, 63, 84$ , which are all indeed sacred. There are therefore 4 sacred numbers.

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.