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Selection tests for the Balkan Mathematical Olympiad, 2013

Algebra Difficulty 7.4 National Olympiad, round 2 Prove it Saudi Arabia

The base-7 representation of number $n$ is $\overline{a b c}_{(7)}$ , and the base-9 representation of number $n$ is $\overline{c b a}_{(9)}$ . What is the decimal (base-10) representation of $n$ ?

Solution

We have $n = c + 7b + 49a = a + 9b + 81c$ . This implies that $20(2c - a) = (4a - b)$ . Because $0 \leq a, b, c \leq 6$ , either $4a - b = 2c - a = 0$ or $4a - b = 20$ and $2c - a = 1$ .

1. If $4a - b = 2c - a = 0$ , then $b = 8c \leq 6$ and therefore $a = b = c = 0$ , that is $n = 0$ .

2. If $4a - b = 20$ and $2c - a = 1$ , then $a = 5$ , or $6$ . If $a = 5$ then $b = 0$ and $c = 3$ and $n = 248$ . If $a = 6$ , then $2c = 7$ , which is impossible.

Therefore, $n = 0$ or $248$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.