Question

Let a, b be two natural numbers and d be their greatest common

divisor.
Show that there exists a pair x, y ∈Zsuch that d = ax + by.

Answer 1

Yes, the pair (x, y) exists and satisfies the given equation.

Yes, there exists a pair of integers (x, y) such that the greatest common divisor (d) of a and b can be expressed as a linear combination of the two numbers, i.e. d = ax + by.

To prove this, first note that the greatest common divisor (d) of a and b divides both a and b. Therefore, by the Division Algorithm, there exist integers q and r such that a = dq + r and 0 ≤ r < d. Similarly, there exist integers p and s such that b = dp + s and 0 ≤ s < d.

Subtracting these two equations yields d = (a - dq) + (b - dp) = (r - q) + (s - p). Therefore, if we let x = r - q and y = s - p, then d = ax + by, where x and y are both integers. Thus, the pair (x, y) exists and satisfies the given equation.