A Great Connection!

So that's it. If a set A containing finite number of integers, not all of which are zero, is selected, then those integers generate an ideal of the integer ring. But, any ideal of the integer ring is generated by a single positive integer a, which must be the smallest positive integer in that particular ideal. Then every integer in A is a multiple of a. Then a is a common divisor of them.

Now, if b is a positive integer which divides all elements of A, then the ideal generated by b is either equal to or includes the ideal generated by the elements of A. In the first case, b is the smallest integer in the ideal and hence is equal to a. In the second case, a is included in the ideal generated by b. Hence, b divides a (and a>b). This shows that a is the greatest common divisor (gcd) of the integers in A.

The integer a can be expressed as a linear combination of the integers in A because a is in the ideal generated by them. That's why the gcd of a finite number of integers can be expressed as a linear combination of those integers.

What a simple but great connection!