9.3. The teacher wrote four different natural numbers in Petya's notebook. For each pair of these numbers, Petya found their greatest common divisor. He got six numbers: 1, 2, 3, 4, 5, and \( N \), where \( N > 5 \). What is the smallest value that \( N \) can have?

(O. Dmitriev)

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# 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 $N \geqslant 14$.

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 $k$. The lemma is proved.

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.

Comment. Only the answer - 0 points.

Only the answer and an example where $N=14-2$ points.

Proved only that $N \leqslant 14-4$ points.

If this is not proved, but it is proved that $N$ is even - 2 points (these points can be combined with points for the answer).