# 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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