The product of the gcd and lcm of two natural numbers is equal to the product of those numbers

The following paragraphs [A] and [B] are the "tecnhical definitions of gcd and lcm:

[A] The greatest common divisor (gcd) of two given natural numbers m,n is a natural number d satisfying the following two conditions:
   (a)   it must be a divisor of both given numbers:   d|m   and  d|n;
   (b) any natural number x that is a divisor of both given numbers must be a divisor of the greatest common divisor:    if   x|m   and  x|n  then  x|d.
Notation: d = gcd(m,n).

[B] The least common multiple (lcm) of two given natural numbers m,n is an natural number k satisfying the following two conditions:
   (a)  it must be a multiple of both given numbers:   m|k   and  n|k;
   (b) any natural number x that is a multiple of both given numbers must be a multiple of the least common multiple:
            if   m|x  and  n|x  then  k|x.
Notation: k = lcm(m,n).

Let d = gcd(m,n)    and    k = lcm(m,n).
The theorem can be expressed as:              dk = m x n .


The proof is in two parts.

First part of the proof:   dk | mn     product of gcd and lcm divides the product of the natural numbers.

By [A] the quotient   n/d   is an natural number. Then the product   m (n/d) = mn/d   is also an natural number, which means
(*)             m | mn/d.
Similarly, by [A] the quotient   m/d    is a natural number. Then the product   n (m/d) = mn/d   is also an natural number and
(**)            n | mn/d
By (*) and (**) the natural number   mn/d   is a common multiple of both m and n. By [B]   least common multiple    must divide all such multiples. Therefore
                    k | mn/d.
By theorem [ ]
(***)             kd |mn.


Second part of the proof:   mn | dk     the product of the natural numbers divides the product of gcd and lcm.

Since mn is a multiple of both m and n, by [B] the least common multiple must be able to divide completely into it, that is, the quotient mn/k is an natural number. According to [B] both quotients    k/m   and k/n    are natural numbers. The product of any two natural numbers is a natural number (closure) so

n = (mn/k) (k/m)       and       m = (mn/k) (k/n)
Therefore      mn/k | n       and       mn/k | m.
This means that the natural number expression   mn/k   is a divisor of both n and m. By [A] it is a divisor of the greatest common divisor, that is mn/k | d. By theorem [ ]
(****)             mn | kd

(***) and (****) show that   kd   and   mn    divide each other. By theorem [ ] they must be equal.


A numerical method showing the above theorem about the product of gcd and lcm

The method of compmlete factorization can be used to find the gcd and lcm of two natural numbers 3528, 111132. The complete factorizations of these two numbers are:
3528 = 23 x 32 x 72      111132 = 22 x 34 x 73
Now choose the smallest exponent of 2, the smallest exponent of 3 and the smallest exponent on 7 to get:
22 x 32 x 72 = gcd(3528,111132) = 1764,         23 x 34 x 73 = lcm(3528,111132) = 222264
The second expression is obtained by selecting the larger of the exponents on the prime factors.

The product of the numbers 3528 and 111132 is equal to 25 x 36 x 75 = 392 073 696. The product of gcd(3528,111132) and lcm(3528,111132) = 25 x 36 x 75 = 1764 x 222264 = 392 073 696.