Extended Euclidean Algorithm: Proof and Examples, Lecture notes of Grammar and Composition

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Extended Euclidean Algorithm

Yan Huang

Can you sell exactly 4 gallons of gasoline?

3 Gallon 8 Gallon

Can you sell 1 gallon of gasoline with

containers of these two sizes?

3 Gallon 8 Gallon

1. Pour 8*2 gallons of gasoline the tank.
2. Fill the 3-gallon container with gasoline in the tank, 5 times.
3. Sell the remaining 1 gallon gasoline in the tank.

Theorem : gcd(𝑎, 𝑏) = 𝑑 if and only if 𝑑 is the least positive integer that can be expressed as 𝑎𝑥 + 𝑏𝑦 where 𝑥, 𝑦 ∈ ℤ.

Theorem : gcd(𝑎, 𝑏) = 𝑑 if and only if 𝑑 is the least positive integer that can be expressed as 𝑎𝑥 + 𝑏𝑦 where 𝑥, 𝑦 ∈ ℤ.

Proof (by contradiction ): Consider the set of integers S = 𝑎𝑥 + 𝑏𝑦 𝑥, 𝑦 ∈ ℤ} and 𝑑 = min 𝑆. Assume (for the sake of contradiction) that 𝑑 ∤ 𝑎. Then 𝑎 = 𝑑𝑞 + 𝑟 where 0 ≤ 𝑟 < 𝑑. Therefore, 𝑟 = 𝑑 − 𝑎𝑞 = 𝑎𝑥 + 𝑏𝑦 − 𝑎𝑞 = 𝑎 𝑥 − 𝑞 + 𝑏𝑦 ∈ 𝑆, which contradicts to the fact that 𝑑 = min 𝑆 since 𝑟 ∈ 𝑆 and 𝑟 < 𝑑. Thus, the assumption was wrong and 𝑑|𝑎. Theorem : gcd(𝑎, 𝑏) = 𝑑 if and only if 𝑑 is the least positive integer that can be expressed as 𝑎𝑥 + 𝑏𝑦 where 𝑥, 𝑦 ∈ ℤ.

Examples

• 𝑎 = 6 and 𝑏 = 8

Examples

• 𝑎 = 3 and 𝑏 = 8

Examples

• Find x, y such that 12x + 8y = gcd(12, 8) egcd (12, 8) egcd :: Int - > Int - > (Int, Int, Int) egcd a 1 = (0, 1, 1) egcd a 0 = (1, 0, a) egcd a b | a < b = let (x, y, d) = egcd b a in (y, x, d) | otherwise = let (x, y, d) = egcd b (a mod b) q = a div b in (y, x-y*q, d)

Exercise

• Find integer x, y such that 27x + 42y = gcd(27, 42) egcd (27, 42)