Greatest Common Divisors Proof

Created at: 2025-05-11

Necessary concepts

Definition of Divisibility For Integers

A nonzero integer a is said to divide an integer b if b/a = k for some integer k. When a divides b, we write a | b. When a does not divide b, we write a † b.

Definition of Greatest Common Divisors

• Let a and b be integers.
• If c | a and c | b then c is a common divisor of a and b.
• The greatest common divisor of a and b is the largest integer d such that,
• d | a and d | b.

This number (d) is denoted gcd(a,b).

Examples (prior to the proof)

• gcd(6, 8) = 2
• gcd(12, 8) = 4
• gcd(6, -8) = 2
• gcd(7, 15) = 1
• gcd(-5, -20) = 5
• gcd(9, 9) = 9

Insights (prior to the proof)

• If a and b are non-zero, then
• Since 1 | x is true for every integer, then
• 1 | a and 1 | b are also true, then
• The integer 1 is always a common divisor, then
• The greatest common divisor will therefore at least be 1.

• The greatest common divisor can never be larger than |a| or |b|, then
• 1 ≤ gcd(a,b) ≤ |a|, and
• 1 ≤ gcd(a,b) ≤ |b|, and therefore
• gcd(a, b) always exists.

• When a = 0 and b = 0, then there's no greatest common divisor as every integer is a divisor of 0.

Theorem Bézout's identity (prior to the proof)

If a and b are positive integers, then there exist integers k and l such that: gcd(a, b) = ak + bl.

Examples

  gcd(12,20) = 4
  gcd(12,20) = 12k + 20l
  gcd(12,20) = 12*2 + 20*-1
  k = 2, l = -1

  gcd(12,20) = 4
  gcd(12,20) = 12k + 20l
  gcd(12,20) = 12*-3 + 20*2
  k = -3, l = 2

In fact there are infinite solutions. But only one suffices for the theorem.