<s>[INST] Sum of digits of $\dfrac{360}{2^n}$ always 9 - proper explanation? What is the exact explanation behind this "series"?


*

*$\dfrac{360} {2^1} = 180 => 1+8+0 = 9$

*$\dfrac{360}{2^2} = 90 => 9+0 = 9$


..


*$\dfrac{360}{2^{10}} => 0.3515625 => 0+3+5+1+5+6+2+5 = 27 => 2+7 = 9$


and so forth..
how is this kind of "recursive series" explained?
 [/INST] Note that multiplying by $10$ doesn't change the sum of the digits, even if the digits extend beyond the decimal point, so the sum digits in $\frac{360}{2^n}$ is the same as the sum of digits of $\frac{360}{2^n} \times 10^n$, which is equal to $360 \times 5^n$. But
$$360 \times 5^n = 9 \times 40 \times 5^n$$
so this number is always divisible by $9$, and you can apply the sum-of-digits trick.

For what it's worth, the same result holds for any number of the form
$$\frac{9k}{2^m \times 5^n}$$
where $k,m,n \in \mathbb{N}$.
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