+1004 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
0.\overline{0121}_3 = 0.01210121 \dots,
which is repeating.
When \frac{1}{288} is expressed in base 25, is it terminating or repeating?
+1953 First, let's convert each number in the fraction to base 25.
1 always stays the same. We want to convert 288 into base 25. We get that
(BD)25=(11×251)+(13×250)=(288)10
So now, we have that 128810=1BD25
Doing some base division, we find that 1BD25=0.¯0.02468ACEGIKN
As shown, the decimals keep repeating, so 1/288 in base 25 is a repeating decimal.
So our answer is repeating.
Thanks! :)
