7. The product of two natural numbers is 68040, and their least common multiple is 3780. Determine these numbers.

## SCHOOL/CITY COMPETITION IN MATHEMATICS
7. Let $m$ and $n$ be the sought natural numbers with $m < n$.

Then $m \cdot n = 68040$ and $\mathrm{V}(m, n) = 3780$.

1 POINT

Since $m \cdot n = \mathrm{D}(m, n) \cdot \mathrm{V}(m, n)$, it follows that $\mathrm{D}(m, n) = 68040 : 3780 = 18$.

2 POINTS

Considering that $18 = 2 \cdot 3 \cdot 3$

and $3780 = 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 5 \cdot 7$,

2 POINTS

the following possibilities exist:

| $m$ | $n$ |
| :--- | :--- |
| $2 \cdot 3 \cdot 3$ | $2 \cdot 3 \cdot 3 \cdot 2 \cdot 3 \cdot 5 \cdot 7$ |
| $2 \cdot 3 \cdot 3 \cdot 2$ | $2 \cdot 3 \cdot 3 \cdot 3 \cdot 5 \cdot 7$ |
| $2 \cdot 3 \cdot 3 \cdot 3$ | $2 \cdot 3 \cdot 3 \cdot 2 \cdot 5 \cdot 7$ |
| $2 \cdot 3 \cdot 3 \cdot 5$ | $2 \cdot 3 \cdot 3 \cdot 2 \cdot 3 \cdot 7$ |
| $2 \cdot 3 \cdot 3 \cdot 7$ | $2 \cdot 3 \cdot 3 \cdot 2 \cdot 3 \cdot 5$ |
| $2 \cdot 3 \cdot 3 \cdot 2 \cdot 3$ | $2 \cdot 3 \cdot 3 \cdot 5 \cdot 7$ |
| $2 \cdot 3 \cdot 3 \cdot 2 \cdot 5$ | $2 \cdot 3 \cdot 3 \cdot 3 \cdot 7$ |
| $2 \cdot 3 \cdot 3 \cdot 2 \cdot 7$ | $2 \cdot 3 \cdot 3 \cdot 3 \cdot 5$ |

The sought numbers are

| $m$ | $n$ |
| ---: | ---: |
| 18 | 3780 |
| 36 | 1890 |
| 54 | 1260 |
| 90 | 756 |
| 126 | 540 |
| 108 | 630 |
| 180 | 378 |
| 252 | 270 |

Note: If not all possibilities are listed, then score as follows:

| NUMBER OF POSSIBILITIES LISTED | NUMBER OF POINTS |
| :---: | :---: |
| 7,6 | 8 |
| 5,4 | 6 |
| 3,2 | 4 |
| 1 | 2 |

## SCHOOL/CITY COMPETITION IN MATHEMATICS