I OM - B - Task 4

Find two natural numbers $ a $ and $ b $ given their greatest common divisor
$ D=12 $ and least common multiple $ M=432 $. Provide a method for finding
solutions in the general case.
If the greatest common divisor of numbers $ a $ and $ b $ is 12, then

where numbers $ x $ and $ y $ are coprime. In this case, the least common multiple of numbers $ 12x $ and $ 12y $ is $ 12xy $, hence

Numbers $ x $ and $ y $ can be found by factoring 36 into a product of two coprime factors.
There are two such factorizations:

We obtain two solutions

The sought numbers are

In the general case, when the greatest common divisor of numbers
$ a $ and $ b $ is $ D $, and the least common multiple is
$ M $, we reason similarly and obtain the equations:

from which

The problem has as many solutions as there are ways to factor the natural number \frac{M}{D} into two coprime factors $ x $ and $ y $. One of the solutions is always the numbers $ D $ and $ M $.