The Greatest Common Divisor -- Number Theory 4

The converse states that if integers a and b are coprime (i.e., their greatest common divisor is 1), then there exist integer coefficients x and y such that ax+by=1. This is provided by Bézout's identity.

Conditions: a and b are coprime integers.; gcd(a,b) = 1.

The proof demonstrates this by showing that d is a linear combination of a and b. Since any common divisor divides both a and b, it must also divide any integer linear combination of them, including d.

Conditions: d is the least positive linear combination of a and b.; c is a common divisor of a and b.

It serves as a certificate because if integer coefficients x and y satisfy ax+by=1, then any common divisor of a and b must divide 1. The only positive integer dividing 1 is 1 itself, proving that the greatest common divisor is 1.

Conditions: a and b are integers.; There exist integers x and y such that ax+by=1.

It contradicts minimality because the remainder can be expressed as a linear combination of a and b. If the remainder is positive and smaller than d, it belongs to the set of positive linear combinations, violating the definition of d as the smallest such value.

Conditions: d is defined as the least positive integer linear combination of a and b.; The remainder r is obtained by dividing a or b by d.; r is positive and $r < d$.

The well-ordering principle guarantees this by ensuring that any nonempty set of positive integers has a smallest element. Since the set of positive linear combinations is nonempty, it must contain a minimum value.

Conditions: Integers a and b are not both zero.; The set consists of positive values of the form au+bv where u and v are integers.