Why is the product of two numbers divided by their greatest common divisor the least common multiple

Why is the product of two numbers divided by their greatest common divisor the least common multiple

The greatest common divisor is the multiplication of all the same prime numbers of two numbers. The least common multiple is to subtract once and multiply all the prime numbers that want to be the same. The complement is just right, so the product of two numbers divided by their greatest common divisor is the least common multiple

The greatest common divisor of two numbers is 20, and the least common multiple is 300. The product of these two numbers is

Because 20 = 2 * 2 * 5
300=2*2*5*5*3
So these two numbers are 100 and 60 or 300 and 20, respectively
So product = 100 × 60 = 6000
300×20=6000

The sum of two numbers is 432. The sum of the greatest common divisor and the least common multiple is 7776. Find the product of two numbers

The greatest common divisor m of two numbers, let two numbers be am, BM, AB coprime, and the least common multiple ABM
AM+BM = (A+B)M = 432=2^4×3^3 …… ①
ABM + M = (AB+1)M = 7776=2^5×3^5 …… ②
Then (2) / 1
(AB+1) / (A+B) = 2*3^2 = 18
AB + 1 = 18A + 18B
If the right of the equation is even, then AB is odd, then a and B must be odd
(B-18)A = 18B - 1
A = (18B - 1)/ (B - 18) = (18B - 324 + 323)/(B-18) = 18 + 323/(B-18)
323 = 17 * 19, divisible by B-18, then
①
B-18 = 17,B = 35,A = 37
or
B-18 = 19,B = 37,A = 35
M = 432 / (35+37) = 6
The two numbers are 35 * 6 and 37 * 6
The product of two numbers = 35 * 6 * 37 * 6 = (7776-6) * 6 = 46620
②
B - 18 = 323,B = 341,A = 1
M = 432 / (323 + 1) is not an integer
To sum up, the two numbers are 35 * 6 = 210 and 37 * 6 = 222, and their product is 46620