2014 AIME II Problems/Problem 4
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