2021 JMPSC Invitationals Problems/Problem 3

Revision as of 15:35, 11 July 2021 by Grisham (talk | contribs) (Solution)

Problem

There are exactly $5$ even positive integers less than or equal to $100$ that are divisible by $x$. What is the sum of all possible positive integer values of $x$?

Solution

$x$ must have exactly 5 even multiples less than $100$. We have two cases, either $x$ is odd or even. If $x$ is even, then $5x < 100 < 6x$. We solve the inequality to find $\frac{50}{3} \leq x \leq 20$, but since $x$ must be an integer we have x = 18, 20. If $x$ is odd, then we can set up the inequality $10x\leq100\leq12x$. Solving for the integers $x$ must be $9$. The sum is $18+20+9$ or $\boxed{47}$

~Grisham

Invalid username
Login to AoPS