So, my bonus question on my test was "Find the 1551th digit of 2^17365". Anyone know how?
The $\displaystyle 155^{\text{th }}$ digit (going left from the 1's digit position) occurs when the exponent(a) as in $\displaystyle 2^a$, exceeds 511.576926.
$\displaystyle 17365_{\text{decimal}} $ is $\displaystyle 100001111010101_{\text{binary}}$
or
$\displaystyle 2^{16384} + 2^{512} + 2^{469} = 2^{17365}$
$\displaystyle 2^{469} $ will NOT affect the value of the $\displaystyle 155^{th}$ digit.
(16384+512) in binary is 100001000000000
dividing that by 512
$\displaystyle \dfrac{(16384+512)}{512} = 33 $
The 155th digit of $\displaystyle 2^{17365}$ is the same as the least significant digit of $\displaystyle 2^{33}$
$\displaystyle 2^{33} = 8589934592$
The $\displaystyle 155^{\text{th}} \text{digit of } 2^{17365}$ is 2
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