Difference between revisions of "2014 AIME II Problems/Problem 4"
(→See also) |
|||
Line 45: | Line 45: | ||
From this, we can see that <math>A=4</math>, <math>B=4</math>, and <math>C=7</math>, so our desired answer is <math>\boxed{447}</math> | From this, we can see that <math>A=4</math>, <math>B=4</math>, and <math>C=7</math>, so our desired answer is <math>\boxed{447}</math> | ||
+ | ==Solution 3== | ||
+ | Noting as above that <math>0.\overline{ab} = \frac{10a + b}{99}</math> and <math>0.\overline{abc} = \frac{100a + 10b + c}{999}</math>, let <math>u = 10a + b</math>. | ||
+ | Then | ||
+ | <cmath>\frac{u}{99} + \frac{10u + c}{999} = \frac{33}{37}</cmath> | ||
+ | |||
+ | <cmath>\frac{u}{11} + \frac{10u + c}{111} = \frac{9\cdot 33}{37}</cmath> | ||
+ | |||
+ | <cmath>\frac{221u + 11c}{11\cdot 111} = \frac{9\cdot 33}{37}</cmath> | ||
+ | |||
+ | <cmath>221u + 11c = \frac{9\cdot 33\cdot 11\cdot 111}{37}</cmath> | ||
+ | |||
+ | <cmath>221u + 11c = 9\cdot 33^2.</cmath> | ||
+ | |||
+ | Solving for <math>c</math> gives | ||
+ | |||
+ | <cmath>c = 3\cdot 9\cdot 33 - \frac{221u}{11}</cmath> | ||
+ | |||
+ | <cmath>c = 891 - \frac{221u}{11}</cmath> | ||
+ | |||
+ | Because <math>c</math> must be integer, it follows that <math>u</math> must be a multiple of <math>11</math> (because <math>221</math> clearly is not). Inspecting the equation, one finds that only <math>u = 44</math> yields a digit <math>c, 7</math>. Thus <math>abc = 10u + c = \boxed{447}.</math> | ||
== See also == | == See also == | ||
{{AIME box|year=2014|n=II|num-b=3|num-a=5}} | {{AIME box|year=2014|n=II|num-b=3|num-a=5}} |
Revision as of 21:26, 25 May 2014
Problem
The repeating decimals and
satisfy
where ,
, and
are (not necessarily distinct) digits. Find the three digit number
.
Solution 1
Notice repeating decimals can be written as the following:
where a,b,c are the digits. Now we plug this back into the original fraction:
Multiply both sides by This helps simplify the right side as well because
:
Dividing both sides by and simplifying gives:
At this point, seeing the factor common to both a and b is crucial to simplify. This is because taking
to both sides results in:
Notice that we arrived to the result by simply dividing
by
and seeing
Okay, now it's pretty clear to divide both sides by
in the modular equation but we have to worry about
being multiple of
Well,
is a multiple of
so clearly,
couldn't be. Also,
Now finally we simplify and get:
But we know is between
and
because it is a digit, so
must be
Now it is straightforward from here to find
and
:
and since a and b are both between and
, we have
. Finally we have the
digit integer
Solution 2
Note that . Also note that the period of
is at most
. Therefore, we only need to worry about the sum
. Adding the two, we get
From this, we can see that
,
, and
, so our desired answer is
Solution 3
Noting as above that and
, let
.
Then
Solving for gives
Because must be integer, it follows that
must be a multiple of
(because
clearly is not). Inspecting the equation, one finds that only
yields a digit
. Thus
See also
2014 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.