Track / Stage 4 / 214 of 340 #474 of 2000

Problem 474

AMC 12 late, AIME early

Number theory Difficulty 4.9 Find the answer

16. You are asked to choose two positive integers, $m$ and $n$ with $m>n$ , so that as many as possible of the expressions $m+n, m-n, m \times n$ and $m \div n$ have values that are prime. When you do this correctly, how many of these four expressions have values that are prime?
A 0
B 1
C 2
D 3
E 4

Your answer

Official solution

16. D All four values cannot be prime. If this were so, both $m \times n$ and $m \div n$ would be prime which can happen only if $m$ is prime and $n=1$ . If $m$ is an odd prime then $m+1$ is even and at least 4 , hence not prime, while if $m=2$ then $m-1$ is not prime but $m+1=3$ is. Thus three prime values are the most we can have.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.