The sum of two different natural numbers is 60, and the sum of their greatest common factor and least common multiple is also 60. How many groups of natural numbers meet the conditions?

The sum of two different natural numbers is 60, and the sum of their greatest common factor and least common multiple is also 60. How many groups of natural numbers meet the conditions?

The sum of two different natural numbers is 60, and the sum of the greatest common factor and the least common multiple is 60
First consider this case, the least common multiple and the greatest common divisor are exactly equal to this. In this case, there is a multiple ratio relationship between the large number and the decimal
For example: (30,30) the greatest common divisor is the same as the least common multiple
For example: (20,40), (15,45), (10,50), (5,55)
You can also factorize 60 to get (12,48) and so on
Therefore, in this case:
60=2*2*3*5
When the factor is 2, there is a group of (30,30)
When the factor is 3, there is a group of (20,40)
When the factor is 4, there is a group of (15,45)!
When the factor is 5, there are (12,48), one group
When the factor is 6, there is a group of (10,50)
When the factor is 10, there is a group of (6,54)
When the factor is 12, there is a group of (5,55)
When the factor is 15, there is a group of (4,56)
When the factor is 20, there is a group of (3,57)
When the factor is 30, there is a group of (2,58)
When the factor is 60, there is a group of (1,59)?
So the total number is C (1,3) + [C (2,4) - C (1,2)] + [C (3,4) - C (2,3)] + C (4,4) = 3 + 4 + 3 + 1 = 11
Then find the value of the greatest common divisor and the least common multiple that are not equal to the two natural numbers! But you can completely prove that there is no such array!