Do you know your bases?

\[169_{a} = 144_{a + 1} = 121_{a + 2} = 100_{a + 3}, \text{ } a \geq 10\]

Albert was experimenting with different bases and noticed a pattern emerging when he compared multiple bases to each other, as shown above.

Without doing any extra work he concludes that the following equation holds true.

\[196_{a} = 169_{a + 1} = 144_{a + 2} = 121_{a + 3} = 100_{a + 4}, \text{ } a \geq 10\]

He shows this work to his friend Bella who claims that it should be the following equation instead.

\[18G_{a} = 169_{a + 1} = 144_{a + 2} = 121_{a + 3} = 100_{a + 4}, \text{ } a \geq 17\]

Albert disagrees.

Who's right? Albert, Bella or both?

Details and Assumptions

  • The letter \(G\) in \(18G_{a}\) is equal to 16.
  • \(n_{a}\) is a number written in base \(a\).
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