Difference between revisions of "2013 AMC 12A Problems/Problem 10"
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<math>\boxed{\textbf{(D)} 143}</math> | <math>\boxed{\textbf{(D)} 143}</math> | ||
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+ | ==Video Solution== | ||
+ | https://www.youtube.com/watch?v=XQpQaomC2tA | ||
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+ | ~sugar_rush | ||
== See also == | == See also == | ||
{{AMC12 box|year=2013|ab=A|num-b=9|num-a=11}} | {{AMC12 box|year=2013|ab=A|num-b=9|num-a=11}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 19:12, 24 November 2020
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,
Solution 3
Notice that we have
We can subtract to get
From this we determine must be a positive factor of
The factors of are and .
For and however, they yield and which doesn't satisfy and being distinct.
For and we have and . (Notice that or can be zero)
The sum of these are
Video Solution
https://www.youtube.com/watch?v=XQpQaomC2tA
~sugar_rush
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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