document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207598>Question 1207598</A>: <I><SPAN STYLE=\"word-break: break-all;\"> When fractions are expressed in different bases, they can be terminating or repeating.  For example, when $\frac{1}{5}$ is expressed in base $3,$ the result is<BR>\n" );
document.write( "\[0.\overline{0121}_3 = 0.01210121 \dots,\]<BR>\n" );
document.write( "which is repeating.\r<BR>\n" );
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document.write( "When $\frac{1}{288}$ is expressed in base $13,$ is it terminating or repeating? </SPAN>\n" );
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document.write( "The fraction 1/n, when expressed in base b, will terminate if and only if the prime factorization of n contains only factors that are prime factors of the base b.<br><BR>\n" );
document.write( "For example, in our familiar base 10 system, since the prime factorization of 10 is 2*5, a fraction 1/n will terminate if and only if the prime factorization of n contains only 2's and 5's.<br><BR>\n" );
document.write( "In your example, the base 13 is a prime number, so 1/n expressed in base 13 will terminate if and only if the prime factorization of n contains only 13's -- i.e., if and only if n is a power of 13.<br><BR>\n" );
document.write( "288 is not a power of 13, so 1/288 expressed in base 13 will not terminate.<br><BR>\n" );
document.write( "ANSWER: Repeating<br><BR>\n" );
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