2009 AIME I Problems/Problem 13
Problem
The terms of the sequence defined by
for
are positive integers. Find the minimum possible value of
.
Solution
Solution 1
This question is guessable but let's prove our answer
let put into
now
and set them equal now
let's rewrite it
Let make it looks nice and let
Since and
are integer, we can see
But we can't have an infinite sequence of proper factors, unless
Thus,
So now, we know
To minimize , we need
and
Thus, answer
Solution 2
If , then either
or
All the integers between and
would be included in the sequence. However the sequence is infinite, so eventually there will be a non-integral term.
So , which
. When
,
. The smallest sum of two factors which have a product of
is
See also
2009 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |