Difference between revisions of "2017 AIME I Problems/Problem 2"
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Revision as of 15:53, 8 March 2017
Problem 2
When each of ,
, and
is divided by the positive integer
, the remainder is always the positive integer
. When each of
,
, and
is divided by the positive integer
, the remainder is always the positive integer
. Find
.
Solution
Let's tackle the first part of the problem first. We can safely assume:
Now, if we subtract two values:
which also equals
Similarly,
Since
is the only common factor, we can assume that
, and through simple division, that
.
Using the same method on the second half:
Then.
The common factor is
, so
and through division,
.
The answer is