Difference between revisions of "1988 USAMO Problems/Problem 1"
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First, split up the nonrepeating parts and the repeating parts of the decimal, so that the nonrepeating parts equal to <math>\frac{a}{b}</math> and the repeating parts of the decimal is equal to <math>\frac{c}{d}</math>. | First, split up the nonrepeating parts and the repeating parts of the decimal, so that the nonrepeating parts equal to <math>\frac{a}{b}</math> and the repeating parts of the decimal is equal to <math>\frac{c}{d}</math>. | ||
Revision as of 18:22, 5 April 2013
Problem
The repeating decimal 
, where 
and 
are relatively prime integers, and there is at least one decimal before the repeating part. Show that is divisble by 2 or 5 (or both). (For example, 
, and 88 is divisible by 2.)
Solution
First, split up the nonrepeating parts and the repeating parts of the decimal, so that the nonrepeating parts equal to 
and the repeating parts of the decimal is equal to 
.
Suppose that the length of 
is 
digits. Then 
Since 
, after reducing the fraction, there MUST be either a factor of 2 or 5 remaining in the denominator. After adding the fractions 
, the simplified denominator will be 
and since 
has a factor of 
or 
, must also have a factor of 2 or 5.
Q.E.D.
See Also
| 1988 USAMO (Problems • Resources) | ||
| Preceded by First Question |
Followed by Problem 2 | |
| 1 • 2 • 3 • 4 • 5 | ||
| All USAMO Problems and Solutions | ||
