Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2021 JMPSC Invitationals Problems/Problem 13"

(Created page with "==Problem== Let <math>p</math> be a prime and <math>n</math> be an odd integer (not necessarily positive) such that <cmath>\dfrac{p^{n+p+2021}}{(p+n)^2}</cmath> is an integer....")
(No difference)

Revision as of 15:08, 11 July 2021

Problem

Let $p$ be a prime and $n$ be an odd integer (not necessarily positive) such that \[\dfrac{p^{n+p+2021}}{(p+n)^2}\] is an integer. Find the sum of all distinct possible values of $p \cdot n$.

Solution

Assume temporarily that $p \neq 2$. Then, $p$ and $n$ are both odd which implies that the numerator is odd and the denominator is even. Since an even number cannot divide an odd number, we have a contradiction. Since our claim is incorrect, $p=2$ and we now wish to make \[\frac{2^{n+2023}}{(n+2)^2}\] an integer. We see that the denominator is always odd, while the numerator is always even. Thus, the only possible values for $n$ are when the denominator is $1,-1$, which implies $n=-1,-3$. These correspond with $p=2$, so $pn=-2,-6$ for an answer of $-8$. ~samrocksnature

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2021_JMPSC_Invitationals_Problems/Problem_13&oldid=157942"