2007 iTest Problems/Problem 53
The following problem is from the Ultimate Question of the 2007 iTest, where solving this problem required the answer of a previous problem. When the problem is rewritten, the T-value is substituted.
Problem
Three distinct positive Fibonacci numbers, all greater than , are in arithmetic progression. Let
be the smallest possible value of their sum. Find the remainder when
is divided by
.
Solution
By definition, for a Fibonacci number, and
. From the definition,
. That means the numbers
,
, and
are in arithmetic progression with common difference
.
Writing out the Fibonacci numbers, the first numbers that come after are
,
,
, and
. That means the desired three numbers are
,
, and
. The sum of the three numbers is
, and the remainder after dividing by
is
.
See Also
2007 iTest (Problems) | ||
Preceded by: Problem 52 |
Followed by: Problem 54 | |
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