# Thread: Find the 155th digit of 2^17365

1. ## Find the 155th digit of 2^17365

So, my bonus question on my test was "Find the 1551th digit of 2^17365". Anyone know how?

2. Originally Posted by ISeriouslyNeedHelp
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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3. Originally Posted by aidan
$\displaystyle 2^{16384} + 2^{512} + 2^{469} = 2^{17365}$
THis is not correct.

4. http://www.wolframalpha.com/input/?i=1551th+digit+of+2^17365

cheat!