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Difference between revisions of "2021 JMPSC Invitationals Problems/Problem 3"

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#[[2021 JMPSC Invitational Answer Key|2021 JMPSC Invitational Answer Key]]
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#[[JMPSC Problems and Solutions|All JMPSC Problems and Solutions]]
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Revision as of 16:26, 11 July 2021

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

See also

  1. Other 2021 JMPSC Invitational Problems
  2. 2021 JMPSC Invitational Answer Key
  3. All JMPSC Problems and Solutions

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