WMC 2018 Pursuit - Pack 2, B1
If for positive integer bases and , what is the smallest value of that is greater than ?
The answer is 13.
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1 solution
The conditions of the problem imply that 6 b + 4 = 4 c + 6 ⟹ 3 b + 2 = 2 c + 3 ⟹ 2 c = 3 b − 1 .
Therefore, 2 c ≡ 2 ( m o d 3 ) ⟹ c ≡ 1 ( m o d 3 ) . The smallest integer greater than 10 that satisfies this is 1 3 .