Difference between revisions of "2002 AMC 12A Problems/Problem 20"
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| Line 19: | Line 19: | ||
The repeating decimal <math>0.\overline{ab}</math> is equal to | The repeating decimal <math>0.\overline{ab}</math> is equal to | ||
<cmath> | <cmath> | ||
| − | \frac{ | + | \frac{10a+b}{100} + \frac{10a+b}{10000} + \cdots |
= | = | ||
| − | + | (10a+b)\cdot\left(\frac 1{10^2} + \frac 1{10^4} + \cdots \right) | |
= | = | ||
| − | + | (10a+b) \cdot \frac 1{99} | |
= | = | ||
| − | \frac{ | + | \frac{10a+b}{99} |
</cmath> | </cmath> | ||
Revision as of 19:30, 28 February 2012
Problem
Suppose that 
and 
are digits, not both nine and not both zero, and the repeating decimal 
is expressed as a fraction in lowest terms. How many different denominators are possible?
Solution
The repeating decimal is equal to
![\[\frac{10a+b}{100} + \frac{10a+b}{10000} + \cdots = (10a+b)\cdot\left(\frac 1{10^2} + \frac 1{10^4} + \cdots \right) = (10a+b) \cdot \frac 1{99} = \frac{10a+b}{99}\]](http://latex.artofproblemsolving.com/7/5/e/75e79d4e1c4a5a5a280b24504155a203de95908f.png)
When expressed in lowest terms, the denominator of this fraction will always be a divisor of the number 
. This gives us the possibilities 
. As and are not both nine and not both zero, the denumerator 
can not be achieved, leaving us with 
possible denominators.
(The other ones are achieved e.g. for 
equal to 
, 
, 
, 
, and , respectively.)
See Also
| 2002 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 19 |
Followed by Problem 21 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
