3. We took four natural numbers. For each pair of these numbers, we wrote down their greatest common divisor. Six numbers were obtained: 1, 2, 3, 4, 5, N, where $N>5$. What is the smallest value that the number $N$ can take? (O. Dmitriev)
Answer: 14. Solution: The number $N$ can equal 14, as shown, for example, by the quartet of numbers 4, $15, 70, 84$. It remains to show that $N \geq 14$.

Lemma. Among the pairwise GCDs of four numbers, there cannot be exactly two numbers divisible by some natural number $k$. Proof. If among the original four numbers there are no more than two numbers divisible by $k$, then among the pairwise GCDs, no more than one is divisible by $k$. If, however, three of the original numbers are divisible by $k$, then all three of their pairwise GCDs are divisible by $k$.

Applying the lemma to $k=2$, we get that the number $N$ is even. Applying it to $k=3, k=4$ and $k=5$, we get that $N$ is not divisible by 3, 4, and 5. Therefore, $N$ cannot equal $6, 8, 10$, and 12.