Difference between revisions of "2013 AMC 12A Problems/Problem 10"
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<math> \textbf{(A)}\ 11\qquad\textbf{(B)}\ 44\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 143\qquad\textbf{(E)}\ 155\qquad </math> | <math> \textbf{(A)}\ 11\qquad\textbf{(B)}\ 44\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 143\qquad\textbf{(E)}\ 155\qquad </math> | ||
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==Solution 1== | ==Solution 1== | ||
Note that <math>\frac{1}{11} = 0.\overline{09}</math>. | Note that <math>\frac{1}{11} = 0.\overline{09}</math>. |
Revision as of 00:16, 10 March 2018
Contents
Problem
Let be the set of positive integers
for which
has the repeating decimal representation
with
and
different digits. What is the sum of the elements of
?
Solution 1
Note that .
Dividing by 3 gives , and dividing by 9 gives
.
The answer must be at least , but cannot be
since no
other than
satisfies the conditions, so the answer is
.
Solution 2
Let us begin by working with the condition . Let
. So,
. In order for this fraction
to be in the form
,
must be a multiple of
. Hence the possibilities of
are
. Checking each of these,
and
. So the only values of
that have distinct
and
are
and
. So,
See also
2013 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 9 |
Followed by Problem 11 |
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 |
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