## Task 1 - 080711

The greatest common divisor of two natural numbers is 6, and their least common multiple is 210.

Determine all pairs of numbers with the given properties!
Both sought numbers are 1. divisors of 210 and 2. multiples of 6. Since $210=2 \cdot 3 \cdot 5 \cdot 7$, the numbers that are both divisors of 210 and multiples of 6 are: 6, 30, 42, and 210 and only these. The following pairs of numbers need to be examined:

(1) 6 and 30

(2) 6 and 42

(3) 6 and 210

(4) 30 and 42

(5) 30 and 210

(6) 42 and 210.

For the first two pairs, the least common multiple is 30 and 42, respectively, but not 210; the pairs 5 and 6 have the greatest common divisor as 30 and 42, respectively, but not 6. For pairs 3 and 4, the conditions stated are met.

The pairs 6 and 210 as well as 30 and 42 are the only ones that satisfy the given conditions.