For on the one hand, a number \(1 \lt d \leq m\) which divides \(n\) serves as a certificate for the membership of \(\langle n,m \rangle\) in \(\sc{FACTORIZATION}\). And on the other hand, in order to demonstrate the membership of \(\langle n,m \rangle\) in \(\overline{\sc{FACTORIZATION}}\), it suffices to exhibit a prime factorization of \(n\) in which no factor is less than \(m\). , falsifying valuations in the case of \(\sc{VALIDITY}\)) and because the primality of the individual factors can