2020 CIME II Problems/Problem 5
Problem
A positive integer is said to be
-consecutive if it can be written as the sum of
consecutive positive integers. Find the number of positive integers less than
that are either
-consecutive or
-consecutive (or both), but not
-consecutive.
Solution
The smallest -consecutive positive integer is
, and every multiple of
greater than
is also
-consecutive, with the last one less than
being
. There are
-consecutive positive integers less than
. The smallest
-consecutive positive integer is
, and the largest one less than
is
. There are
of these. However, we counted
twice and we only wanted to count them once, so we subtract
from our total, giving us a total of
that are either
- or
-consecutive. The ones that are also
-consecutive are
for a total of
integers to be removed. The answer is
.
See also
2020 CIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All CIME Problems and Solutions |
The problems on this page are copyrighted by the MAC's Christmas Mathematics Competitions.