3. Find all pairs of natural numbers $a, b$ for which

$$
n(a, b)+D(a, b)=63
$$

where $n(a, b)$ denotes the least common multiple and $D(a, b)$ the greatest common divisor of the numbers $a, b$.
SOLUTION. We will use what we stated in the 1st problem. Let $a=D p, b=D q, n=D p q$, where $D$ is the greatest common divisor, $n$ is the least common multiple of the numbers $a, b$, and the numbers $p, q$ are coprime. According to the problem statement, $D(1+p q)=63$ must hold, so we have the following options (without loss of generality, we assume that $a \leqq b$):

| $D$ | $p q$ | $(p, q)$ |  | $(a, b)$ |  |
| :---: | :---: | :---: | :---: | :---: | :---: |
| 1 | 62 | $(1,62)$, | $\overline{(2,31)}$ | $(1,62)$, | $\overline{(2,31)}$ |
| 3 | 20 | $(1,20)$, | $(4,5)$ | $(3,60)$, | $(12,15)$ |
| 7 | 8 | $(1,8)$ |  | $(7,56)$ |  |
| 9 | 6 | $(1,6)$, | $(2,3)$ | $(9,54)$ | $(18,27)$ |
| 21 | 2 | $(1,2)$ |  | $(21,42)$ |  |

The problem has 8 solutions, not distinguishing the order of the numbers $a, b$.

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