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I came up with the following exercise with the intention of learning Mathematica but I am already stuck in the beginning. Let $ n,b $ denote positive integers with $ n>0 $ and $ b\geq2 $. My aim is to check whether the base-$ b $ representation of $ n $ contains the same digits in base $ b $.

For instance, $ n=63_{10} $ has the representation $ n=111111_2 $ in base $ b=2 $ and $ n=124_{10} $ is $ n=444_5 $ in base $ b=5 $.

I know that Mathematica can easily convert between two different bases using BaseForm[expr, base], but I have no idea how to efficiently verify whether the digits of the resulting number are all the same. I was looking for functions that work with repunits, but I found none. Any help is appreciated!

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    $\begingroup$ Have a look at IntegerDigits (and, if you should ever need it, its inverse FromDigits). For example, IntegerDigits[63, 2] returns {1,1,1,1,1,1} and IntegerDigits[124, 5] returns {4,4,4}. $\endgroup$ – Henrik Schumacher Dec 17 '18 at 9:05
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As @HenrikSchumacher mentioned, IntegerDigits works:

allSameQ[n_, b_] := Equal @@ IntegerDigits[n, b]

allSameQ @@@ {{63, 2}, {124, 5}}
{True, True}
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  • $\begingroup$ That's perfect, thank you! $\endgroup$ – Klangen Dec 17 '18 at 9:24

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