Difference between revisions of "1988 USAMO Problems/Problem 1"
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Suppose that the length of <math>0.ab\cdots k</math> is <math>p</math> digits. Then <math>\frac{a}{b} = \frac{0.ab\cdots k}{10^{p+1}}</math> Since <math>0.ab\cdots k < 10^{p+1}</math>, after reducing the fraction, there MUST be either a factor of 2 or 5 remaining in the denominator. After adding the fractions <math>\frac{a}{b}+\frac{c}{d}</math>, the simplified denominator <math>n</math> will be <math>\\lcm(b,d)</math> and since <math>b</math> has a factor of <math>2</math> or <math>5</math>, <math>n</math> must also have a factor of 2 or 5. | Suppose that the length of <math>0.ab\cdots k</math> is <math>p</math> digits. Then <math>\frac{a}{b} = \frac{0.ab\cdots k}{10^{p+1}}</math> Since <math>0.ab\cdots k < 10^{p+1}</math>, after reducing the fraction, there MUST be either a factor of 2 or 5 remaining in the denominator. After adding the fractions <math>\frac{a}{b}+\frac{c}{d}</math>, the simplified denominator <math>n</math> will be <math>\\lcm(b,d)</math> and since <math>b</math> has a factor of <math>2</math> or <math>5</math>, <math>n</math> must also have a factor of 2 or 5. | ||
− | + | <math>\blacksquare</math> | |
==See Also== | ==See Also== |
Revision as of 18:27, 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.
See Also
1988 USAMO (Problems • Resources) | ||
Preceded by First Question |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |