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A base-10 integer n can be represented as \(32_a\) in one base and \(23_b\) in another base, where a and b are any integer bases larger than 3. What is the smallest possible sum a+b?

 Jan 14, 2019

\(3a+2 = 2b+3,~a,b \in \mathbb{N}, a,b > 3\\ 3a=2b+1\\ 2b+1 \text{ is odd, so }a \text{ must be odd}\\ a=5,b=7 \text{ will be the smallest valid values of }a,b\\ a+b = 12\)

 Jan 14, 2019

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