Estonian Math Competitions

(a) The pair $24,27$ is interesting, as the greatest prime factor of them both is $3$. However, there are no primes between them.

(b) The number $1$ and prime numbers cannot form a pair with any smaller integers. However, each composite number forms a pair with its greatest prime factor, which is clearly smaller. Among the first $25$ positive integers, the primes are $2,3,5,7,11,13,17,19$ and $23$. Taking into account also the number $1$, there are $10$ numbers among the first $25$ positive integers that don't form an interesting pair with any smaller positive integer.