Difference between revisions of "2021 JMPSC Invitationals Problems/Problem 3"
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==Solution 2== | ==Solution 2== | ||
− | Suppose <math>x</math> is odd. We have <math>xk</math> for <math>k \equiv 0 \mod 2</math> must work for <math>xk \le 100</math>. Clearly <math>k=\{2,4,6,8,10 \}</math>, which means the maximum value that <math>xk</math> can take on is <math>90=9 \cdot 10</math>, and the minimum value it can take on is <math>2=2 \cdot 1</math>. Since we need | + | Suppose <math>x</math> is odd. We have <math>xk</math> for <math>k \equiv 0 \mod 2</math> must work for <math>xk \le 100</math>. Clearly <math>k=\{2,4,6,8,10 \}</math>, which means the maximum value that <math>xk</math> can take on is <math>90=9 \cdot 10</math>, and the minimum value it can take on is <math>2=2 \cdot 1</math>. Since we need <b>exactly 5</b> even integers, only <math>x=9</math> will work. Now, suppose <math>x</math> is even. We have <math>1 \le k \le 5</math>, which means <math>x=\{18,20 \}</math> hold exactly <math>5</math> even integer multiples. The answer is <math>18+20+9=\boxed{47}</math> |
~Geometry285 | ~Geometry285 |
Latest revision as of 20:06, 11 July 2021
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
The problems on this page are copyrighted by the Junior Mathematicians' Problem Solving Competition.