Number Theory Easy Problem

All primes, except the smallest one 2, are odd. Since $r$ is not the smallest prime, $r$ must be odd. For $r$ to be odd, either $p$ or $q$ must be even and the only even prime is 2, the smallest prime. Since $p< q$ , $p = \boxed 2$ .

R must be either an odd number or an even number. If it is an even number, then it must be 2 for it to be prime. But both P and Q are greater than 1, and lower than R, which we assume is 2. There is a contradiction, so R must be an odd number.

Then either P or Q is the even prime number, 2. If Q was 2, P cannot be greater than 1 and less than 2 at the same time, so P is 2 .