2021 JMPSC Invitationals Problems/Problem 3
Contents
[hide]Problem
There are exactly even positive integers less than or equal to
that are divisible by
. What is the sum of all possible positive integer values of
?
Solution
must have exactly 5 even multiples less than
. We have two cases, either
is odd or even. If
is even, then
. We solve the inequality to find
, but since
must be an integer we have x = 18, 20. If
is odd, then we can set up the inequality
. Solving for the integers
must be
. The sum is
or
~Grisham
Solution 2
Suppose is odd. We have
for
must work for
. Clearly
, which means the maximum value that
can take on is
, and the minimum value it can take on is
. Since we need exactly 5 even integers, only
will work. Now, suppose
is even. We have
, which means
hold exactly
even integer multiples. The answer is
~Geometry285
See also
- Other 2021 JMPSC Invitationals Problems
- 2021 JMPSC Invitationals Answer Key
- All JMPSC Problems and Solutions
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