17. The greatest common divisor of two natural numbers is 12, and their least common multiple is 672. What are these numbers if it is known that the smaller of them is divisible by 7, and the larger is not?
17. Let $a$ and $b$ be the sought numbers. Since $D(a, b) = 12$, it follows that $a = 12x$ and $b = 12y$, where $x$ and $y$ are relatively prime natural numbers, i.e., $D(x, y) = 1$.

Given the equality $D(a, b) \cdot V(a, b) = a \cdot b$, we have $12 \cdot 672 = 12x \cdot 12y$, which simplifies to

$12xy = 672$, i.e., $xy = 56$.

We now have the following possibilities:

| $x$ | 1 | 2 | 4 | 7 |
| :---: | :---: | :---: | :---: | :---: |
| $y$ | 56 | 28 | 14 | 8 |
| $D(x, y)$ | 1 | 2 | 2 | 1 |

Since the numbers $x$ and $y$ are relatively prime, and given that the smaller of them is divisible by 7, while the larger is not, we conclude that the only solution is obtained for $x = 7$ and $y = 8$.