document.write( "Question 1207585: What are the first $5$ digits after the decimal point (technically the hexadecimal point...) when the fraction $\frac{2}{17}$ is written in base $16$?\r
\n" ); document.write( "\n" ); document.write( "(Express your answer as a five digit hexadecimal number. You do not need to include the subscript to indicate the base.) \n" ); document.write( "
Algebra.Com's Answer #845548 by greenestamps(13203)\"\" \"About 
You can put this solution on YOUR website!


\n" ); document.write( "Do some long division in base 16 (\"2\" base 16 divided by \"11\" base 16) and verify the result.
\r\n" );
document.write( "\r\n" );
document.write( "      -----------------\r\n" );
document.write( "   11 ) 2. 0 0 0 0 ...\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "           1\r\n" );
document.write( "      -----------------\r\n" );
document.write( "   11 ) 2. 0 0 0 0 ...\r\n" );
document.write( "        1  1\r\n" );
document.write( "        -------------\r\n" );
document.write( "           F 0\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "           1 E\r\n" );
document.write( "      -----------------\r\n" );
document.write( "   11 ) 2. 0 0 0 0 ...\r\n" );
document.write( "        1  1\r\n" );
document.write( "        -------------\r\n" );
document.write( "           F 0\r\n" );
document.write( "           E E\r\n" );
document.write( "           ---------\r\n" );
document.write( "             2 0

\n" ); document.write( "The remainder is the same number we started with, so the digits \"1E\" will repeat.

\n" ); document.write( "ANSWER: 0.1E1E1

\n" ); document.write( "Check to see that this hexadecimal is equal to 2/17:

\n" ); document.write( "0.1E1E1E... base 16 = ...

\n" ); document.write( "\"1%2F16%2B14%2F16%5E2%2B1%2F16%5E3%2B14%2F16%5E4\"+...

\n" ); document.write( "\"%281%2F16%2B14%2F16%5E2%29%2B%281%2F16%5E2%29%281%2F16%2B14%2F16%5E2%29\"+...

\n" ); document.write( "\"%2816%2F16%5E2%2B14%2F16%5E2%29%2B%281%2F256%29%2816%2F16%5E2%2B14%2F16%5E2%29\"+...

\n" ); document.write( "\"%2830%2F16%5E2%29%2B%281%2F256%29%2830%2F16%5E2%29\"+...

\n" ); document.write( "Use the formula for the sum of an infinite geometric sequence...

\n" ); document.write( "
\r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
\n" );